How to transform graph states using single-qubit operations: computational complexity and algorithms

Graph states are ubiquitous in quantum information with diverse applications ranging from quantum network protocols to measurement based quantum computing. Here we consider the question whether one graph (source) state can be transformed into another graph (target) state, using a specific set of quantum operations (LC + LPM + CC): single-qubit Clifford operations (LC), single-qubit Pauli measurements (LPM) and classical communication (CC) between sites holding the individual qubits. This question is of interest for effective routing or state preparation decisions in a quantum network or distributed quantum processor and also in the design of quantum repeater schemes and quantum error-correction codes. We first show that deciding whether a graph state |G⟩ can be transformed into another graph state |G′⟩ using LC + LPM + CC is NP -complete, which was previously not known. We also show that the problem remains NP-complete even if |G′⟩ is restricted to be the GHZ-state. However, we also provide efficient algorithms for two situations of practical interest. Our results make use of the insight that deciding whether a graph state |G⟩ can be transformed to another graph state |G′⟩ is equivalent to a known decision problem in graph theory, namely the problem of deciding whether a graph G′ is a vertex-minor of a graph G. The computational complexity of the vertex-minor problem was prior to this paper an open question in graph theory. We prove that the vertex-minor problem is NP -complete by relating it to a new decision problem on 4-regular graphs which we call the semi-ordered Eulerian tour problem.


Contents
1 Introduction captures how complex the quantum state is. One way to describe quantum states is by a technique called tree-tensor networks and it was shown in [50] that the minimum dimension of the tensors needed to describe a state is in fact given by the Schmidt-rank width.
In the domain of complexity theory, the rank-width and related measures such as the tree and clique-width [16,5] also form a measure of the inherent complexity of the underlying problem, and feature prominently in the study of fixed-parameter tractable (FPT) algorithms [21]. Specifically, a problem is called fixed-parameter tractable in terms of a parameter r, if any instance I of the problem of fixed r, is solvable in time f (r) · |I| O(1) , where |I| is the size of the instance and f is an arbitrary function of r [21]. In this work, the r is the rank-width, and for graphs of constant rank-width the techniques of Courcelle [17] and its generalizations [15], can be used to obtain polynomial time algorithms for problems such as Graph Coloring [24], or Hamiltonian Path [35]. While very appealing from a complexity theory point of view, however, a direct application of these techniques does however not usually lead to polynomial time algorithms that are also efficient in practice, since f (r) is often prohibitively large.
Since the problem of deciding whether a graph state |G is a qubit-minor of |G (QUBITMINOR) is equivalent to deciding if G is a vertex-minor of G (VERTEXMINOR) [19], an efficient algorithm for VERTEXMINOR, directly provide an efficient algorithm for QUBITMINOR. This in turn can be used for fast decisions on how to transform graph states in a quantum network or distributed quantum processor. However, not much was previously known about the computational complexity of VERTEXMINOR and therefore if efficient algorithms exists. For a related but slightly more restrictive minor-relation, namely pivot-minors it has been shown in [18] that checking whether a graph G has a pivot-minor isomorphic to another graph G is NP-Complete. However the complexity of deciding whether G is a vertex-minor of G was left as an open problem. What's more, we emphasize that for our application we are interested in preparing a specific target state G on a specific set of qubits, as qubits are generally not interchangeable in the applications of our algorithm. As such, our question is not whether we can obtain a graph that is isomorphic to G , but rather whether we can prepare G on a specific set of vertices.
Evidently, for fixed rank-width, it is not difficult to apply the techniques of Courcelle [17], to obtain an FPT algorithm for our problem that is efficient if both the size of G , as well as the rank-width of G are bounded (as we have shown in [19]). Indeed, a powerful method for deciding if a graph problem is fixed-parameter tractable is by Courcelle's theorem and its generalizations [15]. It turns out that also for our case, however, a direct implementation of Courcelle's theorem does not give an algorithm that is efficient in practice. In fact, in the case of VERTEXMINOR, this constant factor obtained by applying the techniques of Courcelle in [19] can be shown to be 1 a power of twos f (r) = 2 2 · · · 2 r (1) where r is the rank-width of the input graph G and the height of the tower is 10 [19].

Results
In this paper we determine the computational complexity of VERTEXMINOR and therefore of QUBITMINOR. In particular we prove that it is in general NP-Complete to decide whether a graph G is a vertex-minor of another graph G. We however also give efficient algorithms for this problem whenever the input graphs belong to particular graph classes. Any overview of the complexity of the problem for different classes of graphs considered in this paper can be seen in fig. 2.
We point out that our results of NP-Completeness and the presented algorithms also apply to the more general class of stabilizer states of relevance in quantum error correction. This is because any stabilizer state can be transformed to some graph state using only single-qubit Clifford operations. Furthermore, given a stabilizer state on n qubits, a graph state equivalent under single-qubit Clifford operations can be found efficiently in time O(n 3 ) [49]. 1. We show that the problem of deciding whether a graph G is a vertex-minor of another graph G is NP-Complete. This implies that QUBITMINOR is also NP-Complete. The hardness part of the above completeness result is obtained by considering a less general version of VERTEXMINOR where G is restricted to be a star graph, which corresponds to transforming graph states to GHZ states. We call this problem STARVERTEXMINOR and show that it is also NP-Complete, even when G is in a strict subclass of circle graphs. To show NP-Completeness of STARVERTEXMINOR we introduce the concept of a semi-ordered Eulerian tour (SOET) 2 . This new graphtheoretical structure might be of independent interest. We show that the problem of deciding whether a graph 1 Far greater than the number of atoms in the universe. 2 Pronounced as "suit".
allows for a SOET (which we call the SOET problem) can be reduced to STARVERTEXMINOR. Furthermore, we show that deciding whether a 3-regular graph is Hamiltonian (CubHam) can be reduced to SOET. Since the CubHam has previously been shown to be NP-Complete, this implies that VERTEXMINOR is as well, since we also show that VERTEXMINOR is in NP. As a key technical tool we also introduce a new class of graphs which we call triangular-expanded graphs. These graphs allow us to relate Hamiltonian tours on 3-regular graphs to certain Eulerian tours on 4-regular graphs and in turn to reduce CubHam to SOET.
2. We provide an efficient algorithm for STARVERTEXMINOR, which we prove to be correct if G is distancehereditary, or equivalently if G has rank-width one. The run-time of this algorithm is O(|V (G )||V (G)| 3 ), where V (G) denotes the vertex-set of G. This algorithm can therefore be used to decide how to transform graph states, with Schmidt-rank width one, to GHZ-states using single-qubit Clifford operations, single-qubit Pauli measurements and classical communication. As mentioned above, a more general method to find efficient algorithms for certain graph problems on graphs with bounded rank-width is by using Courcelle's theorem [15]. Compared to the algorithm provided by a direct implementation of Courcelle's theorem, see [19], our algorithm presented here does not suffer from a huge constant factor in the runtime, as in eq. (1). In fact, we have implemented our efficient algorithm and see that it typically takes for example 50 ms to run for the case when |V (G)| = 50 on a standard desktop computer, see fig. 1. Furthermore, our algorithm is still efficient even if the size of G is unbounded.
Distance-hereditary graphs, and therefore graphs with rank-width one, are exactly the graphs that can be reached by adding leaves and performing twin-splits from a graph with one vertex [3]. To prove that our algorithm is correct we also present some new interesting results relating vertex-minors, distance-hereditary graphs and leaves and twins. For example we show that if v is a leaf or a twin in G but not a vertex in G , then G is a vertex-minor of G if and only if G is a vertex-minor of G \ v, where \v denotes vertex-deletion. : Average and maximal observed run-times for two algorithms that check if a star graph on four vertices is a vertex-minor of a randomly generated connected graph G on vertices V . Random connected distance-hereditary graphs are generated by starting from a single-vertex graph and randomly adding leaves or performing twin-splits, see section 2.5.1, which generates any connected distance-hereditary graph [3]. "Our alg." refers to the algorithm described in section 4.1.1 and "Brute" is the non-efficient algorithm described in [19]. The algorithm of [19] based on the techniques of Courcelle [17] is not depicted here since the pre factor makes an application impractical in practice whenever |V | < f (r) of eq. (1). For each size of V , 10 random graphs are generated for "Brute" and 100 random graphs for "Our alg.", from which the average "(avg.)" and max "(max)" runtime is computed. Both algorithms are implemented in SAGE [47] and the tests were performed on an iMac with 3.2 GHz Intel Core i5 processor with 8 GB of 1600 MHz RAM.
3. We call k−STARVERTEXMINOR the restriction of STARVERTEXMINOR where G is restricted to a star graph having k vertices, corresponding to a GHZ-state (up to LC) on k qubits. We show that k−STARVERTEXMINOR is in P if G is a circle graph 3 . This means that STARVERTEXMINOR is fixed-parameter tractable in the size of G on circle graphs. Interestingly the class of circle graphs has unbounded rank-width ([41, Proposition 6.3] and [14]) and the corresponding graph states therefore have unbounded entanglement according to the Schmidt-rank width. 4. We show that any connected graph G on three vertices or less is a vertex-minor of any connected graph G if and only if the vertices of G are also in G. An efficient algorithm for finding the transformation that takes the former graph to the latter is also provided. 5. We also prove several technical results that may be interesting in their own right. An example of this is theorem 4.4 which points out the interesting behavior of a certain class of graphs with respect to a bipartition of their vertices. Another example would be theorem 2.8 where we show that any distance heredity graph one more than four vertices has a foliage (the set of leaves, axils and twins in the graph) of size at least four.

Simple Graphs
Circle Graphs

Distancehereditary NP-Complete
Unbounded rank-width  The definitions of these classes graphs are given in section 2.4 and section 2.5.1. We show that the STARVERTEXMINOR is NP-Complete (result 1.) on graphs in a strict subclass of circle graphs (red). The same therefore holds for any superclass and in particular for circle graphs (dark blue). On the other hand we show that k−STARVERTEXMINOR is in P on circle graphs (result 3.) On distance-hereditary graphs (green) we found an efficient algorithm for STARVERTEXMINOR (result 2.) Finally it is still an open question if k−STARVERTEXMINOR is NP-Complete on simple graphs (light blue) in general.

Overview
The paper is structured as follows. In section 2 we describe graph states and consider several notions of graph theory we will need throughout the paper. We also introduce the VERTEXMINOR and STARVERTEXMINOR problems and the notion of a semi-ordered Eulerian tour. We also prove a few technical results concerning distance hereditary graphs, circle graphs and vertex-minors which we will need later. Result 1. above is given in section 3, result 2. in section 4.1, result 3. in section 4.2 and result 4. in section 5.

Preliminaries
In this section we set our notation and recall various concepts which will be used throughout the rest of the paper. We start by providing the definitions of graph states, qubit-minors and the relation to vertex-minors. We then recall the definitions of local complementation and vertex deletion as operations on graphs and use these to state the problems VERTEXMINOR and STARVERTEXMINOR which are the core objects of study in this paper. Furthermore, we discuss circle graphs and their various characterizations and discuss how local complementation behaves on these graphs. We also introduce the concept of semi-ordered Eulerian tours, which is a key technical concept for the results later in this paper. Finally we discuss distance-hereditary graphs, which form a subclass of circle graphs. We discuss how these graphs can be built up out of elementary pieces and prove some technical results which will be used later.

Notation and definitions
Here we introduce some notation and vocabulary that will be used throughout this paper. We assume familiarity of the general notation of quantum information theory, see [39] for more details.

Quantum operations
The Pauli matrices will be denoted as The single-qubit Clifford group C consists of operations which leave the Pauli group P = iI, X, Z invariant. More formally, C is the normalizer of the Pauli group, i.e.
where U is the single-qubit unitary operations.

Sequences and words
A sequence X = x 1 x 2 . . . x k is an ordered, possibly empty, tuple of elements in some set X. We also call a sequence a word and its elements letters. We write X ⊆ X, when all letters of X are in the set X. A sub-word X of X, is a word which can be obtained from X by iteratively deleting the first or last element of X. We denote the concatenation of two words X 1 = x 1 . . . x k1 and X 2 = y 1 . . . y k2 as X 1 X 2 = x 1 . . . x k1 y 1 . . . y k2 . We also denote the 'mirror image' by an overset tilde, e.g. if X = ab then X = ba.

Sets
The set containing the natural numbers from 1 to n is denoted [n]. The symmetric difference X∆Y between two sets X and Y is the set of elements of X and Y that occur in X or Y exclusively, i.e. X∆Y = (X ∪ Y ) \ (X ∩ Y ).

Graphs
A simple undirected graph G = (V, E) is a set of vertices V and a set of edges E. Edges are 2-element subsets of V for simple undirected graphs. Importantly, we only consider labeled graphs, i.e. we consider a complete graph with vertices {1, 2, 3} to be different from a complete graph with vertices {2, 3, 4}, even though these graphs are isomorphic. The reason for considering labeled graphs is that these will be used to represent graph states on specific qubits, possibly at different physical locations in the case of a quantum network. In a simple undirected graph, there are no multiple edges or self-loops, in contrast with a multi-graph: An undirected multi-graph H = (V, E) is a set of vertices V and a multi-set of edges E. For undirected multi-graphs, edges are unordered pairs of elements in V . We will often write V (G) = V and E(G) = E to mean the vertex-and edge-set of the (multi-)graph G = (V, E).
Next we list some glossary about (multi-)graphs: • If a vertex v ∈ V is an element of an edge e ∈ E, i.e. v ∈ e, then v and e are said to be incident to one another.
• Two vertices which are incident to a common edge are called adjacent.
• The set of all vertices adjacent to a given vertex v in a (multi-)graph G is called the neighborhood N of v. We will sometimes just write N v if it is clear which (multi-)graph is considered.
• The degree of a vertex v is the number of neighbors of v, i.e. |N v |.
• A k-regular (multi-)graph is a (multi-)graph such that every vertex in the (multi-)graph has degree k.
is an alternating sequence of vertices and edges such that e i is incident to v i and v i+1 for i ∈ [k].
• The vertices v 1 and v k+1 are called the ends of W .
• If the ends of a walk are the same vertex, it is called closed.
• A trail is a walk which does not include any edge twice.
• A closed trail is called a tour.
• A path is a walk which does not include any vertex twice, apart from possibly the ends.
• A closed path is called a cycle.
• Two vertices u and v are called connected if there exist a path with u and v as ends.
• A (multi-)graph is called connected if any two vertices are connected in the (multi-)graph.
is a subgraph on a subset V ⊆ V and with the edge-set has strictly more connected components than G • The distance d G (v, w) between two vertices v, w in a (multi-)graph G is equal to the number of edges in the shortest path that connects v and w.
• The complement G C of a graph G = (V, E) is a graph with vertex-set V C = V and edge-set A graph G is assumed to be simple and undirected, unless specified and will be denoted as G, G i , G ,G or similar. A multi-graph is assumed to be undirected, unless specified and will be denoted as H, H i , H ,H or similar. Furthermore, 3-regular simple graphs will be denoted as R, R i , R or similar and 4-regular multi-graphs as F , F i , F or similar. We will denote the complete graph on a set of vertices V as K V and the star graph on a set of vertices V with center c by S V,c . We will often not care about the choice of center, writing S V to mean any choice of star graph on the vertex set V .

Graph states
A graph state is a multi-partite quantum state |G which is described by a graph G, where the vertices of G correspond to the qubits of |G . The graph state is formed by initializing each qubit v ∈ V (G) in the state and for each edge (u, v) ∈ E(G) applying a controlled phase gate between qubits u and v. Importantly, all the controlled phase gates commute and are invariant under changing the control-and target-qubits of the gate. This allows the edges describing these gates to be unordered and undirected. Formally, a graph state |G is given as where C (u,v) Z is a controlled phase gate between qubit u and v, i.e.
and Z v is the Pauli-Z matrix acting on qubit v.
Any graph state is also a stabilizer state [30]. The GHZ states are an important class of stabilizer states given as It is easy to verify that any graph state given by a star or complete graph, i.e. |S V,c or |K V , can be turned into a GHZ state on the qubits V using only single-qubit Clifford operations.
In the next section we discuss local complementations and vertex-deletions on graph states. It turns out that singlequbit Clifford operations (LC), single-qubit Pauli measurements (LPM) and classical communication (CC): LC+LPM+ CC, which take graph states to graph states, can be completely characterized by local complementations and vertexdeletions on the corresponding graphs. More concretely, any sequence of single-qubit Clifford operations, mapping graph states to graph states, can be described as some sequence of local complementations on the corresponding graph. Moreover, measuring qubit v of a graph state |G in the Pauli-X, Pauli-Y or Pauli-Z basis, gives a stabilizer state that is single-qubit Clifford equivalent to |X v (G) , |Y v (G) , |Z v (G) respectively. The operations X v , Y v and Z v are graph operations consisting of sequences of local complementations together with the deletion of vertex v, which we define in definition 2.6. As mentioned the post-measurement state of for example a Pauli-X measurement on qubit v is only single-qubit Clifford equivalent to the graph state |X v (G) . The single-qubit Clifford operations that take the post-measurement state to |X v (G) depend on the outcome of the measurement of the qubit v and act on qubits adjacent to v [30]. This means classical communication is required to announce the measurement result at the vertex v to its neighboring vertices.
In [19] we introduced to notion of a qubit-minor which captures exactly which graph states can be reached from some initial graph state under LC + LPM + CC. Formally we define a qubit-minor as: If |G is a qubit-minor of |G , we denote this as In [19] we have shown that the notion of qubit-minors for graph states is equivalent to the notion of vertex-minors for graphs. We will define and discuss vertex-minors in section 2.3.1, however we formally state the relation between vertex-minors here. For a proof see [19].
Theorem 2.1 (Theorem 2.2 in [19]). Let |G and |G be two graph states such that no vertex in G has degree zero. Then |G is a qubit-minor of |G if and only if G is a vertex-minor of G, i.e.
Note that one can also include the case where G has vertices of degree zero. Let's denote the vertices of G which have degree zero as I. We then have that Theorem 2.1 is very powerful since it allows us to consider graph states under LC + LPM + CC, purely in terms of vertex-minors of graphs. We will therefore in the rest of this paper use the formalism of vertex-minors to study the computational complexity of transforming graph states using LC + LPM + CC and provide efficient algorithms for transforming graph state using LC + LPM + CC.

Local complementations and vertex-deletions
Local complementation is a fundamental operation on graphs [10]. We have the following definition.

Definition 2.2 (Local complementation).
A local complementation τ v is a graph operation specified by a vertex v, taking a graph G to τ v (G) by replacing the induced subgraph on the neighborhood of v, i.e. G[N v ], by its complement. The neighborhood of any vertex u in the graph τ v (G) is therefore given by where ∆ denotes the symmetric difference between two sets. Given a sequence of vertices v = v 1 . . . v k , we denote the induced sequence of local complementations, acting on a graph G, as Below we show a simple example of the action of local complementation on a graph (in particular we consider a local complementation on the vertex labeled 2).
If two graphs G 1 and G 2 are related by a sequence of local complementations, i.e. ∃v : τ v (G 1 ) = G 2 , we call the two graphs LC-equivalent and denote this as G 1 ∼ LC G 2 . Checking whether two graphs are LC-equivalent can be done in time O(n 4 ), where n is the size of the graphs, as shown in [11]. This result was used in [48] to find an efficient algorithm for checking whether two graph states are equivalent under single-qubit Clifford operations, by proving that two graph state are equivalent under single-qubit Cliffords if and only if their corresponding graphs are LC-equivalent.
Notable about local complementation is its action on star and complete graphs. For a vertex set V and c ∈ V we have that τ c (S V,c ) = K V and for any v ∈ V we have τ v (K V ) = S V,v . This means all star graphs on a vertex set V are equivalent to each other under local complementation and also to the complete graph on V . Moreover, no other graph is equivalent to the star or complete graph.
Another operation which we will make use of is the pivot.

Definition 2.3 (Pivot).
A pivot ρ e is a graph operation specified by an edge e = (u, v), taking a graph G to ρ e (G) such that The pivot can simply be specified by an undirected edge since as shown in [8].
It will be useful to be able to specify a pivot simply by a vertex v. We therefore also introduce the following definition: Definition 2.4. The graph operation ρ v is specified by a vertex, taking a graph G to ρ v (G) such that where e v is an edge incident on v chosen in some consistent way. For example we could assume that the vertices of G are ordered and that e v = (v, min(N v )). The specific choice will not matter but importantly e v only depends on G and v, and the same therefore holds for ρ v (G).
Another fundamental operation on a graph is that of vertex-deletion, which relates to measuring a qubit of a graph state in the standard basis [30]. We denote the deletion of vertex v from the graph G as It turns out that given a sequence of local complementations and vertex-deletions, acting on some graph, one can always perform the vertex-deletions at the end of the sequence and arrive at the same graph. This fact follows inductively from the following lemma.
Lemma 2.1. Let G = (V, E) be a graph and v, u ∈ V be vertices such that v = u, then Proof. Note first that it is important that v = u since the operation τ v (G \ u) is otherwise undefined. To prove that the graphs G 1 = τ v (G \ u) and G 2 = τ v (G) \ u are equal, we show that the neighborhoods of any vertex in the graphs are the same, i.e. N The local complementation only changes the neighborhoods for vertices which are adjacent to v, so for any vertex w = u which is not adjacent to v, we have that On the other hand, for a vertex w which is adjacent to v, its neighborhood becomes by the definition of a local complementation.

Vertex-minors
Using the two operations local complementation and vertex-deletion, we can formulate the notion of a vertex-minor of a graph.
and if G is not a vertex-minor of G then G ≮ G.
Vertex-minors were first studied in [8] but by the name of l-reductions. Note that if G 1 and G 2 are two LC-equivalent graphs, then G < G 1 if and only if G < G 2 . It is interesting to consider under which conditions a graph G is a vertex-minor of another graph G. As theorem 2.2 below states, to decide whether G < G it is sufficient to check whether G is LC-equivalent to at least one out of 3 |V (G)|−|V (G )| graphs. To formally state the theorem we introduce the following three operations.
Definition 2.6. The graph operations X v , Y v and Z v , specified with a vertex v, act on a graph G by transforming it to When we need to specify which edge incident on v the pivot of X v acts on, we write X The three graph operations X v , Y v and Z v correspond to how Pauli-X, -Y and -Z measurements act on graph states (as proven in [30]). As mentioned in section 2.2, measuring qubit v of a graph state |G in the Pauli-X, -Y or -Z basis gives a stabilizer state which is single-qubit Clifford equivalent to |X v (G) , |Y v (G) and |Z v (G) respectively. Equations (25) to (27) shows examples of how these operations can act on graphs.
The operation X (u) v is the most complicated one, so we will here quickly describe what happens to a graph when X (u) v is applied. One can check that after the operation X (u) v , the vertex u will have the neighbors that v previously had, except v itself. Furthermore, some edges between vertices in (N v ∪ N u ) \ {u, v} will be complemented, i.e. removed if present or added if not. To know which of these edges gets complemented, let's introduce the following three sets which form a partition of (N v ∪ N u ) \ {u, v}. In eq. (27), these sets are v} gets complemented if and only if w 1 and w 2 belong to different sets of the partition (V vu , V v , V u ). All other edges in the graph, i.e. edges containing a vertex not in N v ∪N v , will be unchanged.
It turns out that the three operations {X v , Y v , Z v } are sufficient to check whether some graph is a vertex-minor of another graph. This is formalized in the following theorem we proved in [19].
Theorem 2.2 (Theorem 3.1 in [19]). Let G and G be two graphs and let u = (v 1 , . . . , v l ), where l = |V (G) \ V (G )| be an ordered tuple such that each element of V (G) \ V (G ) occurs exactly once in u. Furthermore, let P u denote the set of graph operations Then we have that G < G ⇔ ∃P ∈ P u : G ∼ LC P (G).
Note that in [19] we indexed P u simply with the set associated to the word u since the statement is independent of the ordering of the elements of u. A direct corollary of the above theorem is therefore: Let G and G be two graphs. Furthermore, let u and u be two ordered tuples such that each element of V (G) \ V (G ) occurs exactly once in both u and u . Then we have that Proof. This follows directly from theorem 2.2 since both sides in eq. (31) are true if and only if G < G.
Note that theorem 2.2 does not give an efficient method to check if G is a vertex-minor of G, since the set P u is of exponential size for all u. To study this problem further we formally define the vertex-minor problem.
Problem 2.1 (VERTEXMINOR). Given a graph G and a graph G defined on a subset of V (G), decide whether G is a vertex-minor of G.
Note again that we deal with labeled graphs here. We will often consider the special case where G is a star graph S V defined on a subset V of V (G). Remember that a graph state described by a star graph is single-qubit Clifford equivalent to a GHZ-state. Thus checking if S V is a vertex-minor of G is equivalent to checking if |G can be transformed to GHZ-state on the qubits V by only using LC + LPM + CC. We will give this problem a separate name.
Problem 2.2 (STARVERTEXMINOR). Given a graph G and a vertex subset V of V (G), decide whether S V is a vertex-minor of G.
Note that we have not specified which star graph on V we use. This is not ambiguous since all star graphs on V are equivalent under local complementation. In the rest of the text we will often leave the choice of star graph open.

Circle graphs
Here we introduce circle graphs and representations of these under the action of local complementations. Circle graphs are graphs with edges represented as intersections of chords on a circle. These graphs are also sometimes called alternance graphs since they can be described by a double occurrence word such that the edges of the graph are the given by the alternances induced by this word. We will make use of the latter description here, which was introduced by Bouchet in [7] and also described in [12]. This description is also related to yet another way to represent circle graphs, as Eulerian tours of 4-regular multi-graphs, introduced by Kotzig in [34]. For an overview and the history of circle graphs see for example the book by Golumbic [26].

Double occurrence words
Let us first define double occurrence words and equivalence classes of these. This will allow us to define circle graphs.
Definition 2.7 (Double occurrence word). A double occurrence word X is a word with letters in some set V , such that each element in V occur exactly twice in X. Given a double occurrence word X we will write V (X) = V for its set of letters.
Definition 2.8 (Equivalence class of double occurrence words). We say that a double occurrence word Y is equivalent to another X, i.e. Y ∼ X, if Y is equal to X, the mirror X or any cyclic permutation of X or X. We denote by d X = {Y : Y ∼ X} the equivalence class of X, i.e. the set of words equivalent to X.
Next we define alternances of these equivalence classes, which will represent the edges of an alternance graph. Note that if (u, v) is an alternance of d X then so is (v, u), since the mirror of any word in d X is also in d X .
Definition 2.10 (Alternance graph). The alternance graph A(X) of a double occurrence word X is a graph with vertices V (X) and edges given exactly by the alternances of d X , i.e.
Note that since A(X) only depends on the equivalence class of X, the alternance graphs A(X) and A(Y ) are equal if X ∼ Y . Now we can formally define 4 circle graphs.
Definition 2.11 (Circle graph). A graph G which is the alternance graph of some double occurrence word X is called a circle graph.

Eulerian tours on 4-regular multi-graphs
There is yet another way to represent circle graphs, closely related to double occurrence words, as Eulerian tours of 4-regular multi-graphs.
Definition 2.12 (Eulerian tour). Let F be a connected 4-regular multi-graph. An Eulerian tour U on F is a tour that visits each edge in F exactly once.
Any 4-regular multi-graph is Eulerian, i.e. has a Eulerian tour, since each vertex has even degree [6].
Furthermore, any Eulerian tour on a 4-regular multi-graph F traverses each vertex exactly twice, except for the vertex which is both the start and the end of the tour. Such a Eulerian tour induces therefore a double occurrence word, the letters of which are the vertices of F , and consequently a circle graph as described in the following definition.
Definition 2.13 (Induced double occurrence word). Let F be a connected 4-regular multi-graph on k vertices V (F ). Let U be a Eulerian tour on F of the form with x i ∈ V . Note that every element of V occurs exactly twice in U . From a Eulerian tour U as in eq. (33) we define an induced double occurrence word as To denote the alternance graph given by the double occurrence word induced by a Eulerian tour, we will write A(U ) ≡ A(m(U )).
Similarly to double occurrence words, we also introduce equivalence classes of Eulerian tours under cyclic permutation or reversal of the tour.
Definition 2.14 (Equivalence class of Eulerian tours). Let F be a connected 4-regular multi-graph and U be an Eulerian tour on F . We say that an Eulerian tour U on F is equivalent to U , i.e. U ∼ U , if U is equal to U , the reversal U or any cyclic permutation of U or U . We denote by t U the equivalence class of U , i.e. the set of Eulerian tours on F which are equivalent to U .
It is clear that if the Eulerian tours U and U on a 4-regular multi-graph F are equivalent, then so are the double occurrence words m(U ) and m(U ). Furthermore, as for double occurrence words, two equivalent Eulerian tours on a connected 4-regular multi-graph induce the same alternance graph.

Local complementations on circle graphs
We will now introduce an operation τ v on double occurrence words that will be the equivalent of performing a local complementation on the corresponding alternance graph.
. Let X be a double occurrence word and v be an element in V (X). We can then always find sub-words A, B and C not containing v, such that X = AvBvC. Note that some of the sub-words A, B and C are possibly empty. The operation τ v acting on a double occurrence word is then defined as If v = (v 1 , . . . , v l ) is a sequence of elements of V (X) we use the notation τ v (X) = τ v l • · · · • τ v1 (X).
The operation τ v in the above definition maps equivalence classes to equivalence classes, as defined in definition 2.8. That is, if X ∼ Y and v ∈ V (X), then τ v (X) ∼ τ v (Y ). For example, assume that Y is the mirror of X, i.e. Y = X.
Then we see that The case when Y is a obtained by a cyclic permutation of X can be checked similarly.
In [12] it was shown that the alternance graph of A( τ v (X), where X is a double occurrence word and v ∈ V (X), is the same as the graph obtained by performing a local complementation at v, i.e.
Similar to the above we can also define an operationτ v on Eulerian tours U on 4-regular multi-graphs which also has the effect of a local complementation on the graph A(U ).
Definition 2.16 (τ v ). Let F be a connected 4-regular multi-graph. Let U be a Eulerian tour on F and v be a vertex in V . Let P v be the first subtrail of U that starts and ends at v, i.e. U = U 1 P v U 2 , from some U 1 and U 2 . We definē τ v (U ) to be the Eulerian tour obtained by traversing U 1 , the reversal of P v and then Note in particular thatτ v (U ), where U is an Eulerian tour on F , is also a Eulerian tour on F .
We have now defined τ -operations on circle graphs, τ -operations on double occurrence words andτ -operations on Eulerian tours of 4-regular multi-graphs. They are given similar names since they are in some sense the same operation but in different representations of circle graphs. To see this note that where U = U 1 P v U 2 as in definition 2.16. From eq. (37) and the shorthand A(U ) = A(m(U )) we also have that The operationτ v on Eulerian tours of 4-regular multi-graphs was introduced by Kotzig in [33], where he called it a κ-transformation.
As stated by Bouchet in [12], Kotzig [33] proved that any two Eulerian tours of a 4-regular multi-graph are related by a sequence of κ-transformations.
Theorem 2.3 (Proposition 4.1 in [12], [33]). Let U and U be Eulerian tours on the same connected 4-regular multigraph. Then there exist a sequence v such that τ v (U ) ∼ U .

Vertex-deletion on circle graphs
When we are considering vertex-minors of circle graphs, it is useful to have an operation on the double occurrence word that corresponds to the deletion of a vertex in the corresponding alternance graph. Let X = AvBvC be a double occurrence word and v be an element in V (X). We will denote by X \ v the deletion of the element v, i.e.
The resulting word ABC is also a double occurrence word and furthermore we have that If W = {w 1 , w 2 . . . , w l } is a subset of V , we will write X \ W as the deletion of all elements in W , i.e.
Connected to this we can also define an induced double occurrence sub-word m[W ] = X \ (V \ W ). The reason for calling this an induced double occurrence sub-word stems from its relation to induced subgraphs of the alternance graph as

Vertex-minors of circle graphs
Since we now have expressions for local complementation and vertex deletion on circle graphs in terms of double occurrence words, we can consider vertex-minors of circle graphs completely in terms of double occurrence words. More precisely we have the following theorem.
Theorem 2.4. Let G be a circle graph such that G = A(X) for some double occurrence word X. Then G is a vertex-minor of G if and only if there exist a sequence v of elements in V (G) = V (X) such that Proof. By using eq. (43) and eq. (39) on the right hand side of eq. (44) we have that Since G is a vertex-minor of G = A(X) if and only if there exist a sequence v of elements in V (G) such that the theorem follows.
We can also consider vertex minors of circle graphs in terms of their representations as Eulerian tours on connected 4-regular multi-graphs, which we will use in section 3 to prove that VERTEXMINOR is NP-Complete. Theorem 2.3, together with eq. (39), implies that connected 4-regular multi-graphs describe equivalence classes of circle graphs under local complementations. Bouchet pointed out this fact in [12]. We formalize this here as a theorem together with a formal proof: Theorem 2.5. Let U be an Eulerian tour of a connected 4-regular multi-graph F with vertices V . Then (1) any graph LC-equivalent to A(U ) is an alternance graph of some Eulerian tour of F and (2) any alternance graph of a Eulerian tour of F is a graph LC-equivalent to A(U ).
Proof. We start by proving (1), so let us therefore assume that G is a graph LC-equivalent to A(U ). This means, by definition, that there exist a sequence v of vertices in V such that G = τ v (A(U )). By using eq. (39) we have that which shows that G is an alternance graph induced by a Eulerian tour of F , sinceτ v (U ) is a Eulerian tour on F . To prove (2), assume that U is a Eulerian tour of F . We will now prove that the alternance graph of U , A(U ), is LC-equivalent to A(U ). By theorem 2.3, we know that there exist a sequence ofτ v -transformations that relates U and U , i.e. there exist a sequence v such thatτ Since these Eulerian tours are equivalent, their induced alternance graphs are equal, i.e.
Finally, using eq. (39) on the above equation gives which shows that A(U ) and A(U ) are indeed LC-equivalent.
Similarly to theorem 2.4 we can decide if a circle graph has a certain vertex-minor by considering Eulerian tours of a 4-regular graph, which is captured in the following theorem.
Theorem 2.6. Let F be a connected 4-regular multi-graph and let G be a circle graph such that A(U ) for some Eulerian tour U on F . Then G is a vertex-minor of G if and only if there exist a Eulerian tour U on F such that Proof. Lets first assume that G is a vertex-minor of G. This means that there exists a sequence v such that where we used eq. (39) in the first step and eq. (43) in the second. We therefore see that U =τ v (U ) is a Eulerian tour on F satisfying eq. (51).
To prove the converse let us assume that there exist a Eulerian tour U on F satisfying eq. (51). From theorem 2.3 we know that there exist a sequence v such that U =τ v (U ). We can then replace U byτ v (U ) in eq. (51) such that where we again made use of eq. (43) and eq. (39). From eq. (57) we see that G is indeed a vertex-minor of G, see definition 2.5, since G = A(U ).

Semi-Ordered Eulerian tours
From the previous sections we have seen that circle graphs and their vertex-minors can be described by Eulerian tours on connected 4-regular multi-graphs. As discussed in section 2.2, the question of whether a given graph G has vertex-minors on the subset V ⊆ V (G) in the form of star or complete graphs is of importance in quantum information theory, since it corresponds to transforming the graph state |G to a GHZ-state on the qubits V . A natural question is therefore: Given a set of vertices V , what property should a connected 4-regular multi-graph F satisfy, such that S V is a vertex-minor of A(U ), for some Eulerian tour U on F . As we will see in this section, a necessary and sufficient condition is that F allows for what we call a semi-ordered Eulerian tour (SOET) with respect to V .
The existence of a SOET on a four regular graph F with respect to some vertex set V will therefore be a key technical tool when considering STARVERTEXMINOR on circle graphs, as described in section 3. We formally define a SOET as follows.
Definition 2.17 (SOET). Let F be a 4-regular multi-graph and let V ⊆ V (F ) be a subset of its vertices. Furthermore, let s = s 1 s 2 . . . s k be a word with letters in V such that each element of V occurs exactly once in s and where k = |V |. A semi-ordered Eulerian tour U with respect to V is a Eulerian tour such that m(U ) = X 0 s 1 X 1 s 2 . . . s k X k s 1 Y 1 s 2 . . . s k Y k for some s and where X 0 , X 1 , . . . , X k , Y 1 , . . . , Y k are words (possibly empty) with letters in V \ V . This can also be stated as m(U )[V ] = ss, for some s.
Note that the multi-graph F is not assumed to be simple, so multi-edges and self-loops are allowed. A SOET is a Eulerian tour on F that traverses the elements of V in some order once and then again in the same order. The particular order in which the SOET traverses V will not be important here, only that it traverses V in the same order twice. An example of a graph that allows for a SOET with respect to the set V = {a, b, c, d} can be seen in fig We also formally define the SOET-decision problem, which takes a 4-regular multi-graph F and a subset V of the vertices as input and asks to decide whether or not the graph F allows for a Semi-Ordered Eulerian Tour with respect to the vertex set V . Problem 2.3 (SOET). Let F be a 4-regular multi-graph and let V be a subset V (F ). Decide whether there exists a SOET U on F with respect to the set V .
As mentioned, the reason for introducing the notion of a SOET is that a 4-regular multi-graph F allows for a SOET with respect to a subset V ⊆ V (F ) if and only if a star graph on V is a vertex-minor of an alternance graph A(U ) induced by a Eulerian tour U on F . This is captured in the following theorem, formulated as a corollary of theorem 2.6.
Corollary 2.6.1. Let F be a connected 4-regular multi-graph and let G be a circle graph given by the alternance graph of a Eulerian tour U on F , i.e. G = A(U ). Then S V is a vertex-minor of G if and only if F allows for a SOET (see definition 2.17) with respect to V .
Proof. Note first that S V ≤ G if and only if K V ≤ G, since S V and K V are LC equivalent. From theorem 2.6 we know that K V is a vertex-minor of G if and only if there exist an Eulerian tour U on F such that It is easy to verify that A( One can see that the existence of a SOET on a 4-regular multi-graph F with respect to V , imparts an ordering on the subset of vertices V . We will in particular be interested in vertices in V that are 'consecutive' with respect to the SOET. Consecutiveness is defined as follows.
We also define the notion of a "maximal sub-word" associated with two consecutive vertices.
, we call X i and Y i the two maximal sub-words associated with the consecutive vertices s i and s i+1 . Furthermore, we call X k and Y k X 0 the two maximal sub-words associated with the consecutive vertices s k and s 1 . Given two consecutive vertices u and v, we will denote their two maximal sub-words as X and X , Y and Y or similar.

Leaves, twins and axils
In this section we will consider certain vertices called leaves, twins and axils. First we will prove that such vertices can in many cases be removed when considering the vertex-minor problem, which can simplify the problem significantly. We capture this in theorem 2.7. This motivates us to consider distance-hereditary graphs, since it turns out that these are exactly the graphs that can be reached from a single-vertex graph by adding leaves or performing twin-splittings. We will leverage these properties in section 4.1 to find an efficient algorithm for STARVERTEXMINOR when the input graph is distance hereditary. We define and consider distance-hereditary graphs in section 2.5.1.
Let us first formally define leaves, twins and axils.

Definition 2.20 (Leaves and axils).
A leaf is vertex with degree one. An axil is the unique neighbor of a leaf.

Definition 2.21 (Twin).
A twin is a vertex v such that there exist a different vertex u with the same neighborhood, i.e. v is a twin if and only if ∃u ∈ V \ {v} : A vertex u as in eq. (59) is called a twin-partner of v and v, u form a twin-pair. If v and u are adjacent, they form a true twin-pair and otherwise a false twin-pair.
The foliage of a graph G is the set of leaves, axils and twins in a graph G and is denoted We are now ready to prove the following theorem which can be used to simplify some instances of VERTEXMINOR, in particular when considering distance-hereditary graphs, see section 2.5.1.
Theorem 2.7. Let G and G be graphs and v be a vertex in G but not in G . Then the following is true: This means we only need to prove the other direction. Assume therefore that G is a vertex-minor of G. We start by proving the case where v is a leaf in G. The cases where v is an axil or a twin in G then follow by a short argument.
. Furthermore, let u be a sequence of vertices such that each element of V \ V occurs exactly once in u. Since G is a vertex-minor of G, we know by theorem 2.2 that there exists some sequence of operations P ∈ P u , such that P (G) ∼ LC G . Let us denote the i-th operation in P as P (i) , such that P = P (n−k) • · · · • P (1) , where n = |G| and k = |G |. Remember that each operation P (i) deletes the i-th vertex of u from the graph. Furthermore, let's denote the sequence of operations from i through j in P as By corollary 2.2.1 we know that such a P exist for all orderings u of the vertices in V \ V . Without loss of generality we can assume that v is the first element in u. This means that We will now treat all three these cases separately.
Then it is easy to see that G is also a vertex-minor of G \ v, since If P (1) is X v then the axil of v cannot be in V \ V , since the operation X v on a leaf disconnects the axil from its neighbors. Lets denote the axil of v by w and assume again w.l.o.g. that the ordering of V \ V is such that w is the second element of u. Since w is a disconnected vertex after P (1) , any of the three operations {X w , Y w , Z w } act the same, i.e. deleting w. So the action of X v followed by P (2) ∈ {X w , Y w , Z w } is the same as deleting both v and w or in other words It is again clear that G is then a vertex-minor of G \ v, since with a satisfying sequence taking G \ v to an LC-equivalent graph of G being (Z w , P (3) , . . . , P (n−k) ). This proves the theorem when v is a leaf. Now assume that v is a twin in G. To prove that the theorem also hold for twins, we first show that a twin can always be transformed into a leaf by local complementations. Assume that v and w are false twins, and denote one of their common neighbors as n. 5 Then the graphG = τ w • τ n (G) is a graph where v is a leaf and w is an axil. Since LC-equivalent graphs have the same vertex-minors, G is also a vertex-minor ofG. From what we showed above and that v is a leaf, G is also a vertex-minor ofG \ v. Finally, G is then also a vertex-minor of where we used lemma 2.
1. An almost identical argument can be made for the case where v and w are true twins by considering the graphG = τ w (G).
Since by assumption G < G, we know that G <G and from the cases of leaves we have that also G <G \ v, since v is a leaf inG. This completes the proof.

Distance-hereditary graphs
In this section we introduce distance-hereditary graphs. As shown by Bouchet in [9], distance-hereditary graphs are exactly the graphs with rank-width one. These graphs have nice properties which we make use of in section 4.1.

Definition 2.23 (Distance-hereditary). A graph G is distance-hereditary if and only if, for each connected induced subgraph G[A]
and for any two vertices u, v ∈ A the distance between u and v is the same in G and in G The simplest example of a graph which is not distance-hereditary is the five-cycle C 5 . To see this pick two vertices which have distance two in C 5 and denote their unique common neighbor by v. The distance between the same vertices in the connected induced subgraph C 5 [V \ v] is three and thus not the same as in C 5 . It turns out that distance-hereditary graphs are exactly the graphs which do not contain a vertex-minor isomorphic to C 5 [12]. We also note that distance-hereditary graphs form a strict subclass of circle graphs [12].
In [3] an equivalent property of distance-hereditary is shown: A graph is distance-hereditary if and only if it can be obtained from a single-vertex graph using the following three operations: • Add a leaf : Let u be a vertex in a graph G. Add the vertex v and the edge (u, v) to G.
• False twin-split: Let u be a vertex in a graph G. Add the vertex v and the edges {(v, x) : x ∈ N u }.
• True twin-split: Let u be a vertex in a graph G. Add the vertex v and the edges Note that this implies that a distance-hereditary graph always has at least one leaf or twin, i.e. the foliage is nonempty. This fact will be a critical element of the algorithm presented in section 4.1 In the rest of this section we prove some properties of the foliage for distance-hereditary graphs, which we make use of in section 4.1 to find an efficient algorithm for VERTEXMINOR on distance-hereditary graphs. First we show that the twin relation is in fact transitive. This is a technical lemma we will use in later theorems.
Lemma 2.2. Let G be a graph and let u be a vertex of G that is a twin and has twin-partners {t 1 , t 2 , . . . , t k }. Then all vertices in u ∪ {t 1 , t 2 , . . . , t k } are pairwise twins.
Proof. Since u and t i form a twin-pair, for i ∈ {1, 2, . . . , k}, we have that which implies that Thus, we have that This shows that, for i = j, t i and t j form a twin-pair.
Next we prove that adding leaves to a graph G or performing (true or false) twin-splits never decreases the size of the foliage T (G).

Lemma 2.3.
Assume G is a connected, distance-hereditary graph. Let G be a graph formed by doing a twin-split on G or adding a leaf to G. Then where T (G) is the foliage of G.
Proof. To prove this, let us first consider the case when |G| ≤ 2. Since G is connected it is necessary the case that G = K 1 or G = K 2 : Let's now consider the case when |G| > 2. We consider the two cases when G is formed by adding a leaf and performing a twin-split separately: • Assume G is formed by adding a leaf v to G, making u an axil of G .Note first that if u / ∈ T (G), then |T (G)| can only increase since no vertex in T (G) was affected. Let's therefore assume that u ∈ T (G). There are then three possibilities: (1) u is a leaf, (2) u is an axil but not a twin and (3) u is a twin. We consider these three cases separately: Then the axil of u in G, is not in T (G ), but both u and v are. Therefore |T (G)| = |T (G )|.
-(2) Assume u is an axil but not a twin in G. Then u is also an axil in G and we have that |T (G )| = |T (G)| = 1.
* Assume there is only one twin-pair containing u in G. Then the twin-partner of u in G, is not in T (G ), but both u and v are. Therefore |T (G )| = |T (G)|.
* Assume there is more than one twin-pair containing u in G. Then the twin-partners of u are all pairwise twins, by lemma 2.2, and will still be in G . Therefore |T (G )| = |T (G)| + 1.
• Assume G is formed by twin-splitting u in G, creating v and making v and u a twin-pair. Note first that if u / ∈ T (G), then |T (G)| can only increase since no vertex in T (G) was affected. Let's therefore assume that u ∈ T (G). There are then three possibilities: (1) u is a leaf, (2) u is an axil but not a twin and (3) u is a twin. We consider these three cases separately: Then the axil of u in G is either still an axil in G or not, depending on if u and v are true or false twins. In either case, |T (G )| ≥ |T (G)| since v ∈ T (G ).
-(2) Assume u is an axil but not a twin in G. Note that all leaves with u as an axil in G are also twins. These vertices are also twins in G since they are all now also adjacent to v. Thus, |T (G )| = |T (G)| + 1.
-(3) Assume u is a twin in G.
* Assume there is only one twin-pair in G containing u. Then the twin-partner of u in G, may or may not still be a twin-partner of u in G depending on whether the considered twin-pairs are true or false. Therefore the size of T (G) either remains the same or increases by one since again v ∈ T (G ).
* Assume there are more than one twin-pair in G containing u. Then the twin-partners of u are all pairwise twins, by lemma 2.2, and will still be in G . Therefore |T (G )| = |T (G)| + 1.
We now make use of the above theorem to prove that the foliage has a certain minimum size.
Proof. First we explicitly check that the graphs on 2, 3 and 4 vertices has T (G) = 2, T (G) = 3 and T (G) = 4, respectively. 6 Then by lemma 2.3 and the fact that all distance-hereditary graphs can be built up by twin-splits and adding leaves [9], the result follows.
We point out that the theorem does not hold for k > 4. Consider for example a path graph P k on more than four vertices. It is easy to see that size of the foliage in this case is |P k | = 4.
Finally we show that an interesting property regarding the foliage, in relation to cut-vertices. 7 Proof. Pick an arbitrary connected component G i with vertices V i . If G i is just a single vertex, then this vertex is necessarily a leaf in G and is therefore in T (G). Now assume that |G i | > 1, then by using theorem 2.8, we have that there exist at least one twin-pair not containing v or a leaf which is not This proves the corollary.

Complexity
In this section we will consider the time-complexity of VERTEXMINOR. In particular we will show that even a highly restrictive version of the vertex-minor problem is NP-Hard, namely when G is a star graph and G is in a strict subclass of circle graphs. Since we also prove that VERTEXMINOR is in NP this then proves that VERTEXMINOR is NP-Complete.

VERTEXMINOR is in NP
We begin by arguing that the vertex-minor problem is in NP. Given graphs G and G such that G < G, a witness to this relation would be a sequence of local complementations and vertex deletions that takes G to G . It is not a priori clear that this sequence is polynomial in length w.r.t. to the number of vertices of G. However from theorem 2.2 one can argue that whenever there is such a sequence, there is also a sequence of polynomial length. This leads to the following theorem.
Theorem 3.1. The decision problem VERTEXMINOR is in NP.
Proof. Let G and G be graphs, on n and k vertices respectively. Furthermore, let u be a sequence such that each element of V (G) \ V (G ) occur exactly once in u. If G < G then, by theorem 2.2, there exists a sequence of operations P ∈ P u , as specified in eq. (29), such that P (G) ∼ LC G . Furthermore, the sequence of operations P consists of O(n − k) local complementations and vertex-deletions. A witness to the instance (G, G ) will then be the sequence of operations P . On the other hand, if G ≮ G, then by theorem 2.2, there exist no P ∈ P u such that P (G) ∼ LC G .
Given (G, G ) and a sequence of operations P , a verifier can therefore perform the following protocol to check if (G, G ) is a yes-instance of STARVERTEXMINOR.
2. Decide if P (G) ∼ LC G using Bouchet's algorithm for checking if two graphs are LC-equivalent [11].
3. Output yes if Bouchet's algorithm outputs TRUE and no otherwise.
The verifier will therefore output yes if P is such that P (G) ∼ LC G and no if G ≮ G, since then P (G) LC G for any P . Computing P (G) can be done in time O(n 2 (n − k)), since each local complementation can be performed in time O(n 2 ) [11]. Furthermore, checking whether P (G) and G are LC-equivalent can be done in time O(k 4 ) using Bouchet's algorithm [11]. Thus the verifier will output yes or no in time O(n 2 (n − k)) + O(k 4 ).

VERTEXMINOR is NP-Complete
Next we will argue that the problem VERTEXMINOR is also NP-Hard and hence that it is NP-Complete. We will do this through a sequence of three reductions.
• First we will reduce STARVERTEXMINOR to VERTEXMINOR. This is done in theorem 3.2.
• Secondly we will reduce a new problem, which we call the SOET problem, for Semi-Ordered Eulerian Tour, to STARVERTEXMINOR. This is done in section 3.2.1 • Finally we will reduce the problem of deciding whether a 3-regular (or cubic) graph has a Hamiltonian cycle or not (CubHam), which is a known NP-complete problem [25], to the SOET problem. This is the most complicated part of the reduction and is done in several steps in section 3.2.2.
The reductions between these problems are summarized in fig. 4. Eventually we will have the following theorem, which can be considered the main theorem of this section.
VM SVM CubHam SOET k-DPP The dashed line from STARVERTEXMINOR to SOET is restricted to circle graphs. The dashed line from SOET to CubHam is restricted to triangleexpanded graphs Λ(R) where the SOET is with respect to V (R). Finally the dashed line from SOET to k−DPP is for SOET problems where the SOET is with respect to a set of size k.

Theorem 3.2. VERTEXMINOR is NP-Complete
Proof. Note first that STARVERTEXMINOR trivially reduces to VERTEXMINOR. This is so since every yes(no)instance of STARVERTEXMINOR is also an yes(no)instance of VERTEXMINOR. From theorem 3.3 we see that we can reduce the SOET problem to STARVERTEXMINOR and finally from corollary 3.4.1 we see that we can reduce CubHam to the SOET problem. Since CubHam is a known NP-Complete problem this implies that VERTEXMINOR is NP-Hard. From theorem 3.1 we have that VERTEXMINOR is in NP and hence it is NP-Complete. Now we will detail every step in the above reduction. We begin with proving that the SOET decision problem reduces to STARVERTEXMINOR.

Reducing the SOET problem to STARVERTEXMINOR
In this section we show that the SOET problem reduces to STARVERTEXMINOR. For this we will make use of the properties of circle graphs, discussed in section 2. In corollary 2.6.1 we showed that a 4-regular multi-graph F allows for a SOET with respect to a subset of its vertices V ⊆ V (F ) if and only if an alternance graph A(U ) (which is a circle graph), induced by some Eulerian tour on F , has S V as a vertex-minor.
Since circle graphs are a subset of all simple graphs we can then decide whether a 4-regular graph F allows for a SOET with respect to some subset V of it's vertices by constructing the circle graph induced by an Eulerian tour on F and checking whether it has a star-vertex minor on the vertex set V . This leads to the following theorem. Proof. Let (F, V ) be an instance of SOET, where F is a 4-regular multi-graph and V a subset of the vertex set of F . Also let G be a circle graph induced by any Eulerian tour U on F . From corollary 2.6.1 we see that G has S V as a vertex-minor if and only if F allows for a SOET with respect to the vertex set V . Since an Eulerian tour U can be found in polynomial time [23] and since G can be efficiently constructed given U , this concludes the reduction.

Reducing CubHam to the SOET problem
In this section we will prove that the SOET problem, as defined in problem 2.3, is NP-Complete by reducing the problem of deciding if a 3-regular graph is Hamiltonian (CubHam), a well-known NP-Complete problem [25], to the SOET problem (it is in NP by theorem 3.3 and theorem 3.1). For completeness we include the definition of a Hamiltonian graph. We can use this to formally define the CubHam problem. The reduction of CubHam to the SOET problem is done by going though the following steps.
1. Introduce the notion of a (4-regular) triangular-expansion Λ(R) of a 3-regular graph. This is done in definition 3.2.
2. Argue that given a 3-regular graph R, its triangular-expansion can be constructed efficiently. This is done in lemma 3.1.
3. Introduce the notions of skip and true skip that capture an essential behavior of SOETs on triangular-expansions of 3-regular graphs. This is done in section 3.2.4.

4.
Prove that if a 3-regular graph R is Hamiltonian then the triangular-expansion Λ(R) of R allows for a SOET with respect to the set V (R). This is done in lemma 3.3.

5.
Prove that if the triangular-expansion Λ(R) of a 3-regular graph R allows for a special kind of SOET, called a HAMSOET with respect to R (HAMSOETs are defined in definition 3.4, but can be thought of as SOETs with no true skips), then the 3-regular graph R is Hamiltonian. This is done in lemma 3.4.
6. Prove that if the triangular-expansion Λ(R) of a 3-regular graph R allows for a SOET with respect to V (R) then it also allows for a HAMSOET with respect to R. This is done in lemma 3.5.
Performing all these steps will lead to the following theorems. Proof. Let R be a 3-regular graph and let Λ(R) be its triangular-expansion as defined in definition 3.2. If R is Hamiltonian then lemma 3.3 guarantees that Λ(R) allows for a SOET with respect to the vertices V (R). In the other direction, if Λ(R) allows for a SOET with respect to the vertices V (R) then we can see from lemma 3.4 that it also allows for a HAMSOET. The existence of a HAMSOET on Λ(R) then implies, via lemma 3.4 that R has a Hamiltonian cycle and hence that it is Hamiltonian. This proves the theorem.
Proof. The Hamiltonian cycle problem (CubHam) is NP-Complete on 3-regular graphs [25]. We will reduce this problem to the SOET problem. Let R be an instance of CubHam, i.e. a 3-regular graph. From this 3-regular graph we can construct its triangular-expansion Λ(R). In lemma 3.1 it is argued that this construction can be performed in O(|V (R)|) time. We can then use theorem 3.4 to see that R is Hamiltonian if and only if Λ(R) allows for a SOET with respect to the vertex set V (R). Hence there exists an efficient reduction of CubHam to the SOET problem. This means that the SOET problem is NP-Hard. Furthermore, the SOET problem is in NP since it can be efficiently reduced to STARVERTEXMINOR by theorem 3.3, which is in NP by theorem 3.1. Hence the SOET problem is NP-Complete.
The SOET problem is NP-Complete on graphs which are triangular-expansions of planar 3-regular triply-connected graphs, i.e. graphs in the set
Proof. The proof is the same as the proof of corollary 3.4.1 but using the fact that CubHam is NP-Complete on planar triply-connected graphs.

triangular-expansions
It now remains to prove lemmas 3.3 to 3.5. These lemmas will relate Hamiltonian cycles on 3-regular graphs and SOETs on 4-regular multi-graphs by using a mapping from 3-regular graphs to 4-regular multi-graphs. We call this mapping 'triangular-expansion'. We have the following definition.
Definition 3.2 (Triangular-expansion). Let R be a 3-regular graph. A triangular-expansion Λ(R) of a 3-regular graph R is constructed from R by performing the following two steps: where x, y and z are the neighbors of v. We will denote this triangle subgraph associated to the vertex v with

Double every edge that is incident on two subgraphs
The graph Λ(R) will be called a triangular-expansion of R. A multi-graph F that is the triangular-expansion of some 3-regular graph R will also be referred to as a triangular-expanded graph. Note that the triangular-expansion is not uniquely defined, since for each vertex v ∈ R there is a choice how to orient the triangle with respect to the neighbors of v. Furthermore, the number of vertices in Λ(R) is 6 · |V (R)| and the number of edges is 2 · |E(R)| + 9 · |V (R)|. In fig. 5 we show an example of a 3-regular graph and its triangular-expansion.
For a given triangle subgraph T v in a triangular-expanded graph, we will refer to the vertices adjacent to other triangle subgraphs T x , T y , T z as 'outer vertices' and label them according to the triangle subgraph they are adjacent to. Concretely we label the vertex in T v that is adjacent to T w as v (w) , the index signifies which triangle subgraph it connects to. In the following lemma we argue that this construction can be made efficiently in the size of R.
Lemma 3.1. Let R be a 3-regular graph. We can construct its triangular-expansion in O(|V (R)| 2 ) time.
Proof. Let R be a 3-regular graph. Without loss of generality assume some labeling on the vertices of R, i.e.
We begin by constructing the vertex set V (Λ(R)) off the triangular-expansion Λ(R) of R.
and v j , v j , vĵ are the three unique neighbors of v i in R. Constructing this set takes O(|V (R)|) as we must search the set of edges E(R) of R to find the neighbors of v i and we have that This multi-set can be constructed in constant time. Hence the multi-set It is easy to check that the multi-graph defined by the vertex set Λ(R) and edge multi-set E(Λ(R)) is indeed the triangular-expansion of R. This completes the lemma.

Skips and true skips
A key insight in the behavior of the SOET problem on triangular-expanded graphs is the notion of skips. The word skip stems from the fact that since any SOET U on the triangular-expansion Λ(R) of a 3-regular graph is a Eulerian tour, it must traverse any triangle subgraph T v of Λ(R) exactly three times. However in order for U to be a valid SOET with respect to V (R) it must traverse the vertex v exactly two of those three times. This means it must skip the vertex v exactly once while traversing T v .
We state a more formal definition of a (true) skip in terms of maximal sub-words (see definition 2.19).
. Let R be a 3-regular graph and let Λ(R) be its triangular-expansion. Let U be a SOET on Λ(R) with respect to V (R). Let X be a maximal sub-word of m(U ) (definition 2.19) associated to vertices u, v ∈ V (R). We say the sub-trail described by X makes a skip at a vertex w ∈ 2 w (x2) are sub-words of X (up to reflection), where x 1 , x 2 ∈ V (R). Furthermore, if x 1 = x 2 then we say that the trail described by X makes a true skip at w or sometimes that T w contains a true skip.
Note that since X is a maximal sub-word associated to u, v and w / ∈ {u, v}, w cannot be a letter of X. As stated above, there is always exactly one maximal sub-word describing a sub-trail of a SOET that makes a skip at a certain triangle subgraph, as formalized in the following lemma. One can think of this lemma as giving necessary conditions for the existence of a SOET with respect to V (R) on the triangular-expansion of a 3-regular graph R Lemma 3.2. Let R be a 3-regular graph and let Λ(R) be its triangular-expansion. Let U be a Eulerian tour on Λ(R). Let w ∈ V (R) and let T w be its triangle subgraph in Λ(R). If U is a SOET on Λ(R) with respect to V (R) then here exist exactly one maximal sub-trail of U that makes a skip at w.
Proof. We will prove this by showing that there are exactly three maximal sub-trails of U that traverse vertices in T w and that exactly one of these makes a skip at w. Note first that the Eulerian tour U will enter and exit the triangle subgraph T w exactly three times, since there are six edges incident to T w . Hence there exists exactly three distinct edge-disjoint sub-trails, t 1 , t 2 and t 3 of U that exit and enter T w , i.e. the last vertices they traverse in T w will be x (w) i , where x i for i ∈ [3] are the neighbors of w in R. Note that t 1 , t 2 and t 3 each contain at least one vertex in T w and they jointly traverse all edges in T w (Since U is a Eulerian tour). Now consider the vertex w. The Eulerian tour U traverses this vertex exactly twice. There are now two options for the trails t 1 , t 2 , t 3 . Either (1) one of the trails contains the vertex w exactly twice or (2) there are exactly two trails that contain the vertex w exactly once. Now assume U is a SOET with respect to V (R). If option (1) is true the tour U traverses the vertex w twice in succession before traversing any other vertex in the set V . This is in contradiction with the assumption that U is a SOET. Hence if U is a SOET we must have that option (2) is true, that is, exactly two of the sub-trails t 1 , t 2 and t 3 must contain w.
Lets assume without loss of generality that w ∈ t 2 and w ∈ t 3 . Hence we have that w ∈ t 1 and thus t 1 induces a sub-word m(t 1 ) not containing w. The word m(t 1 ) can be extended to a maximal sub-word X not associated to w but describing a sub-trail traversing vertices in T w . Therefore by definition 3.3 X describes a sub-trail making a skip at w. Furthermore, m(t 2 ) has an overlap with two maximal sub-words associated with y 1 , w and w, y 2 , respectively for some y 1 , y 2 ∈ V (R). Similarly for m(t 3 ) and the vertices z 1 , w and w, z 2 . Thus, the maximal sub-words that have an overlap with m(t 2 ) or m(t 3 ) do not make a skip at w.
Finally, there is no other maximal sub-word describing a sub-trail that traverses a vertex in T w . The reason for this is that t 1 , t 2 and t 3 jointly traverse all edges in T w , so any maximal sub-trail traversing a vertex in T w must have an overlap with t 1 , t 2 or t 3 . The lemma then follows since we found exactly one maximal sub-word describing a sub-trail of U making a skip at w, i.e. the unique one that has an overlap with t 1 .

Equivalence between SOET's and Hamiltonian cycles
Now that we have defined the triangular-expansion Λ(R) of a 3-regular graph R and discussed skips we can finally make the central argument of the reduction given in corollary 3.4.1. We begin by proving that if a 3-regular graph is Hamiltonian then its corresponding triangular-expansion Λ(R) allows for a SOET w.r.t. the vertex set V (R). We have the following lemma. Lemma 3.3. Let R be a 3-regular graph and Λ(R) be a triangular-expansion as defined in definition 3.2. If R is Hamiltonian, then Λ(R) allows for a SOET with respect to V (R).
Proof. Let R be Hamiltonian. This means there exists a Hamiltonian cycle M on R. We will prove that there exists a SOET U with respect to V (R) on the triangular-expansion Λ(R) of R by constructing, from the cycle M on R, a tour U that visits every vertex v ∈ V (R) twice in the same order. We will then argue that this tour can always be lifted to a Eulerian tour and hence can be made into a SOET.
Note first that M induces an ordering on the vertices of R which without of loss of generality we will take to be v 1 , . . . v k where k = |V (R)|. Note that for all i ∈ [k − 1] the vertices v i and v i+1 are adjacent in R and so are v k and v 1 . Now consider, for each i ∈ [k − 1] the triangle subgraphs T vi , T vi+1 and Tv in the triangular-expansion Λ(R) of R wherev i is the unique vertex adjacent to v i in R that is not v i+1 or v i−1 . There are now three cases, depending on the orientation of the triangle subgraph T vi . Either (1) in T vi , we can define two edge-disjoint trails on the triangularexpansion Λ(R) by their description as words X i , X i on the vertices of Λ(R).
An illustration of the trails described by these words is given in fig. 6a. We can similarly define two edge-disjoint trails for the cases (2) and (3). We will abuse notation and refer to the words as X i and X i in all three cases. Importantly, for all three these cases the trails described by X i , X i are edge-disjoint and both begin at v . We can extend this definition to trails U 0 , U 0 also for the case of the adjacent vertices v k and v 1 . Note now that the walk U described by the word W = X 0 X 1 . . . X k−1 X 0 X 1 . . . X k−1 is a tour and moreover that it traverses the vertices in V (R) twice in the order v 1 , . . . v k .
The tour U described by the word W above visits every vertex in However it is not yet a Eulerian tour, since it has not traversed all edges in Λ(R). Note that the edges not traversed by U are precisely the triangular-expansions of the edges in R not traversed by the Hamiltonian cycle M . We can easily construct a Eulerian tour out of U by looping over all elements of the word X and whenever we encounter a vertex v , wherev i is defined as above, we check whether U already traverses the edges (v ). If so we continue the loop and if not we insert the trail (v into U at this position. This procedure is illustrated in fig. 6b. Now U is Eulerian and hence a SOET. This completes the proof. Figure 6: The trails used in the proof of lemma 3.3 to construct a SOET on a triangularexpanded graph Λ(R) from a Hamiltonian cycle on R. Figure 6a shows the trails described by the words X i (solid red) and X i (dashed green), defined in eqs. (81) and (82). Figure 6b shows how these trails can be extended to form a Eulerian tour and therefore a SOET with respect to V (R). The dotted gray lines show edges of Λ(R) which are not used by the trails.
Next we define a special type of SOET on triangular-expanded graphs, which we call HAMSOETs. These special SOETs on a triangular-expanded graph Λ(R), are closely related to Hamiltonian cycles on R.
Definition 3.4 (HAMSOET). Let R be a 3-regular graph and Λ(R) its triangular-expansion. Furthermore, let U be a SOET on Λ(R) with respect to V (R). U is called a HAMSOET with respect to R if, for all vertices u, v ∈ V (R) we have that if u and v are consecutive with respect to the SOET U they are also adjacent in the graph R.
Next we prove that if the triangular-expansion of a 3-regular graph R allows for a HAMSOET with respect to R then the 3-regular graph R is Hamiltonian. Hence the tour on R described by s 1 . . . s k visits each vertex in R exactly once and is hence a Hamiltonian cycle.
We would like to prove that for any 3-regular graph R the existence of a SOET on its triangular-expansion Λ(R) implies the existence of a Hamiltonian cycle on R. So far we have proven this only when Λ(R) allows for a HAMSOET. However, a priori not all SOETs on triangular-expansions Λ(R) have to be HAMSOETs. The reason for this is that consecutive vertices in a SOET are not necessarily adjacent in R, since a SOET can contain true skips.
In the following lemma we will prove that consecutive vertices in a SOET are actually adjacent in R, except for two special cases. These special cases can be remedied since we show that for these cases we can always find a different SOET which is actually a HAMSOET.
The arguments proven below are understood the easiest when one reproduces the visual aids given in figs. 8 and 9 as one follows the arguments. We have the following lemma.
Lemma 3.5. Let R be a 3-regular graph and Λ(R) be its triangular-expansion. If Λ(R) allows for a SOET with respect to V (R), then Λ(R) allows for a HAMSOET with respect to V (R).
Proof. To prove this lemma we will go through the following steps Step 1 Note that if two vertices u, v in V (R) (R being a 3-regular graph) are not adjacent in R but are consecutive in a SOET U with respect to V (R) on Λ(R), then the sub-trails of U described by the maximal sub-words associated to u and v must make a non-zero number of true skips in Λ(R) Step 2 Argue by contradiction that this non-zero number of true skips can never be greater than one. This argument leverages lemma 3.6 which states that if a sub-trail of a SOET U with respect to V (R) makes true skips at triangle subgraphs T w1 , T w2 for w 1 , w 2 ∈ V (R) and w 1 , w 2 are adjacent in R then w 1 , w 2 must actually be consecutive in the SOET U and lemma 3.7 which, in a slightly different initial situation than lemma 3.6, also concludes that two vertices must be consecutive in a SOET U with respect to V (R).
Step 3 Argue by contradiction that there are only two possible ways for the sub-trails described by the maximal sub-words associated to u and v to make one true skip each. This argument also leverages lemmas 3.6 and 3.7.
Step 4 Argue that if a triangular-expansion of a 3-regular graph R allows for a SOET U as in Step 3, then it also allows for a SOET U that is a HAMSOET.

Details of step 1
Let R be such that Λ(R) allows for a SOET with respect to V (R). Let U be any such SOET. If U is also a HAMSOET then we are directly done, therefore assume that U is not a HAMSOET. Since U is not a HAMSOET there must exist at least two vertices u, v ∈ V (R) which are consecutive in U , but not adjacent in R. By definition, since u and v are consecutive and U is a SOET, there must exist exactly two different maximal sub-words X, X of m(U ), associated to u and v. Since u and v are not adjacent in R, the trails described by X, X must each traverse vertices in at least one (but possibly more) triangle subgraph that is not T u or T v . Since X and X are maximal sub-words, they can not contain any vertex in V (R) as a letter and hence the sub-trails described by X, X must each make true skips (see definition 3.3) at at least one triangle subgraph.
Assume w.l.o.g. that the sub-trail described by X makes true skips at the triangle subgraphs T s1 , T s2 , . . . T sr , in this order, and similarly for X and T s 1 , . . . T s r . The true skips that the sub-trail described by X makes, must be at different triangle subgraphs than the ones for X , since there can only be exactly one skip per triangle subgraph, due to lemma 3.2. The triangle subgraphs T s1 , T s2 , . . . T sr , T s 1 , . . . T s r are therefore pairwise different. We will call this situation a rr -skip, which is illustrated in fig. 7. Note that by assumption, r and r are both greater than zero. We will first show that the case where either r or r are greater than one can never occur if U is a SOET. Also shown are the sub-trails, of a SOET U , described by maximal sub-words X and X associated to vertices u, v making true skips at triangle subgraphs T s1 , . . . , T sr and T s 1 , . . . , T s r respectively. These sub-trails always begin and end at a vertex in V (R) but their path inside triangle subgraphs is not shown explicitly. Dashed lines are used to indicate that the sub-trails also traverse unspecified further parts of the graph Λ(R).

Details of step 2
As a visual aid for the following argument, refer to fig. 8. We will make an argument by contradiction. Therefore let r be strictly greater than one. Consider the sub-trail described by the sub-word X defined above. By assumption this sub-trail makes true skips at at least two adjacent triangle subgraphs. Take these to be T s1 and T s2 . As we will prove in lemma 3.6, if two adjacent triangle subgraphs T s1 , T s2 contain true skips, then s 1 and s 2 must be consecutive in the SOET U and that there is some maximal sub-word Y of m(U ) associated to s 1 , s 2 that contains the sub-word s This implies the sub-trail described by Y must make a nonzero number of true skips at triangle subgraphs Tŝ 1 , . . . , Tŝ k in that order. Now consider the triangle subgraphs T s1 and Tŝ 1 . As we will prove below, lemma 3.7 applied to the triangle subgraphs T s1 , Tŝ 1 implies that the vertices s 1 andŝ 1 must be consecutive in U . Call the maximal sub-word of m(U ) associated to these vertices W 0 . Moreover, by lemma 3.7 this maximal sub-word contains the sub-word s (ŝ1) 1ŝ (s1) 1 . Similarly we can see that the vertices s 2 andŝ k must be consecutive in U . Call the maximal sub-word of m(U ) associated to these vertices W k . By lemma 3.7 this maximal sub-word contains the sub-word s (ŝ k+1 ) 2ŝ (s2) k+1 . By applying lemma 3.6 again to all triangle subgraph pairs Tŝ i , Tŝ i+1 for i ∈ [k − 1] we come to the conclusion that s i ,ŝ i+1 for i ∈ [k − 1] must be consecutive in U . Call the maximal sub-word of m(U ) associated to these vertices W i . By lemma 3.7 these maximal sub-words contain the sub-wordŝ i+1 . This implies that U must traverse the vertices s 1 ,ŝ 1 , . . . ,ŝ k , s 2 , s 1 ,ŝ 1 , . . . ,ŝ k , s 2 in order. Since by construction {s 1 , s 2 ,ŝ 1 , . . . ,ŝ k } = V (R) we have that U is not a valid SOET on V (R) (this can be seen by noting that e.g. T s 1 already contains a true skip, hence s 1 can not be part of the set). Hence by contradiction we must have r ≤ 1. We can make the same argument for r yielding r ≤ 1.  shown. Also shown are sub-trail of a SOET U labeled by the maximal sub-words that describe them. These sub-trails always begin and end at a vertex in V (R) but their path inside triangle subgraphs is not shown explicitly. Dashed lines are used to indicate that the sub-trails also traverse unspecified further parts of the graph Λ(R). In this argument a contradiction is arrived at by first assuming the that the SOET U has a sub-trail described by the maximal sub-word X associated to the vertices u, v makes true skips at triangle subgraphs T s1 , T s2 . Using lemma 3.6 it is shown that the maximal sub-word Y must exist (associated to vertices s 1 , s 2 ). This then means that the maximal sub-word Y must exist (also associated to s 1 , s 2 ). The sub-trail described by this sub-word must make true skips at triangle subgraphs T w1 , . . . , T w k . Using lemma 3.6 and lemma 3.7 it is then concluded that the SOET U must visit the vertices s 1 , w 1 , . . . w k , s 2 , s 1 consecutively which means U is not a valid SOET (since {s 1 , w 1 , . . . , w k , s 2 } = V (R)). This is a contradiction.

Details of step 3
As a visual aid for the following argument, refer to fig. 9. Now let r = r = 1. This means the sub-trails described by the maximal sub-words X and X associated to the vertices u and v make exactly one true skip each at triangle subgraphs triangle subgraphs T s1 , T s 1 respectively. We will now argue that there are essentially only two ways that a SOET U with the above properties can exist on Λ(R). This argument will again go in steps: Step 3.1 Argue that the SOET U must have a maximal sub-word associated to the vertex s 1 and some other vertex x that describes a trail traversing the edge connecting the triangle subgraphs T u and T s1 .
Step 3.2 Argue that this sub-trail must make a true skip at the triangle subgraph T u . Note that this is equivalent to arguing x = u.
Step 3.3 Apply the same sequence of arguments for the vertices s 1 and v and also for the vertices s 1 and u and s 1 and v.
Step 3.4 Conclude from the fact that the SOET U can only make a single true skip at the triangle subgraphs T u and T v that s 1 and s 1 must be consecutive in U and moreover that the maximal sub-words Z, Z associated to s 1 and s 2 must make true skips at T u and T v respectively.

Details of step 3.1
Consider the second edge connecting the triangle subgraphs T u and T s1 . This is the edge (u (s1) , s (u) 1 ). Since the SOET U is Eulerian it must traverse this edge. Note also that T s1 already contains a true skip (made by the sub-trail described by X). This means, by lemma 3.2 that any sub-trail of U crossing the edge (u (s1) , s (u) 1 ) connecting T u , T s1 must traverse the vertex s 1 . Let Z be the maximal sub-word describing the sub-trail starting at s 1 and containing the sub-word u (s1) s (u) 1 . This maximal sub-word is (by definition) associated to two vertices in V (R). One of these vertices is s 1 and will label the other one x.
We will now argue that x = u and hence that the sub-trail described by Z makes a true skip at T u . We do this by contradiction in the following argument.

Details of step 3.2
For this part of the argument assume that x = u. This means that u is consecutive to both s 1 and v. This means there must be some other maximal sub-word Z associated to s 1 and u. Note that this sub-word cannot contain the sub-word u (s1) s (u) 1 as it is already contained in the maximal sub-words Z and X . Now consider the unique third vertex that is adjacent to u in R (the vertex adjacent to u which is not s 1 or s 1 ). Let us label this vertex w.
Since T s 1 already contains a true skip (which implies Z cannot connect to u by making a true skip at T s 1 ) the sub-trail described by the maximal sub-word Z must make a true skip at w. Now consider the maximal sub-word Y of m(U ) that that describes a sub-trail traversing the unused edge between T s 1 and T u , i.e. Y contains the sub-word u (s 1 ) s (u) shown. Also shown are sub-trail of a SOET U labeled by the maximal sub-words that describe them. These sub-trails always begin and end at a vertex in V (R) but their path inside triangle subgraphs is not shown explicitly. Dashed lines are used to indicate that the sub-trails also traverse unspecified further parts of the graph Λ(R). In Step 3.2 of lemma 3.5 it is argued that if the subtrail described by the maximal sub-word X associated to vertices u, v makes a true skip at the triangle subgraph T s1 then the sub-trail described by the maximal sub-word Z associated to s 1 and another vertex x must make a true skip at the triangle subgraph T u (or equivalently that x = u). This is done by contradiction so it is assumed (as seen in both (a) and (b)) that x = u. This implies the existence of the maximal sub-word Z which must make a true skip at T w where w is the unique neighbor of u s.t. w = s 1 , s 1 . This in turn implies that s 1 and w must be consecutive and connected by a sub-trail described by the maximal sub-word labeled Y in both (a) and (b). This means another sub-trail connecting s 1 and w must exist, which described by a maximal sub-word Y . This sub-word makes true skips at triangle subgraphsŝ 1 , . . .ŝ k . There are now 2 options. Either, as shown in (a) we have that v =ŝ 1 or we have, as shown in (b) that v =ŝ 1 . In the first case (a) lemma 3.7 is applied to conclude that v must be consecutive to s 1 , s 1 and u leading to a contradiction. In the second case (b) lemma 3.6 and lemma 3.7 are used to show that any SOET U must traverse the vertices s 1 , w,ŝ k , . . . ,ŝ 1 , s 1 in order leading to a contradiction.
There are now two possibilities. Either (1) we have thatŝ 1 = v (see fig. 9a) or (2) thatŝ 1 = v (see fig. 9b). We will now consider both of these cases: 1. Ifŝ 1 = v we can apply lemma 3.7 to the vertices v and s 1 to conclude thatŝ 1 = v implies that v and s 1 are consecutive. Call the maximal sub-word connecting them W . We now have that the vertices u, v and s 1 are pairwise consecutive to each other. Since {u, v, s 1 } is a strict subset of V (R), U cannot be a a SOET which is a contradiction.
2. Now assume that v =ŝ 1 . By lemma 3.7 we can now conclude that w andŝ k must be consecutive and that s 1 andŝ 1 = v must be consecutive. We have to perform one last construction to prove the lemma. This construction is visualized in fig. 9b. Call the maximal sub-words associated to these vertex pairs w andŝ k and s 1 andŝ 1 W k and W 0 respectively. We can again use lemma 3.6 to conclude thatŝ i is consecutive toŝ i+1 for all i ∈ [k − 1]. Call the maximal sub-words associated to the verticesŝ i andŝ i+1 W i . This implies that U must traverse the vertices s 1 ,ŝ 1 , . . . ,ŝ k , w, s 1 ,ŝ 1 , . . . ,ŝ k , w in order. Since by construction {s 1 , s w ,ŝ 1 , . . . ,ŝ k } = V (R) we have that U is not a valid SOET on V (R) (this can be seen by noting that e.g. T u already contains a true skip, hence u can not be part of the set).
Hence in both cases we arrive at a contradiction. This means by contradiction that x = u and thus that the sub-trail described by Z makes a true skip at T u .

Details of step 3.3
Now similarly to Step 3.1 consider the edge connecting T s1 and T v . We can again argue that there must exist a maximal sub-word Z associated to the vertex s 1 and some other vertex x that describes a sub-trail that traverses this edge. By the same argument as Step 3.2 we can conclude that x = v and thus that Z describes a sub-trail making a true skip at T v .
We can make the same argument for the vertex s 1 establishing the existence of maximal sub-wordsẐ,Ẑ that describe sub-trails making true skips at T u and T v respectively.

Details of step 3.4
Note that the sub-trails described by Z andẐ make true skips at the triangle subgraph T u . Since T u , by lemma 3.2 can only contain a single true skip and since Z,Ẑ are maximal sub-words we must have that Z ∼Ẑ and thus that the vertices s 1 and s 1 are consecutive with respect to the SOET U . However since we must also conclude that Z ∼Ẑ we have that s 1 and s 1 are consecutive and have two maximal sub-words associated to them. Since there are no further constraints on U imposed by the the fact that the sub-words X, X associated to the vertices v, u make true skips. Therefore SOETs with this type of behavior are in fact allowed. These SOETs can also be found explicitly, as can be sen in fig. 10. If a SOET U has this type of behavior (u and v are not adjacent in R but are consecutive in the SOET U on Λ(R)) we say that the SOET U has a valid 11-skip. Next we show that if a SOET U has a valid 11-skip, and thus is not a HAMSOET, it can always be turned into a HAMSOET by applying a fixed set of local complementation.

Details of step 4
We now show that a SOET with valid 11-skips can be turned into a HAMSOET. The two possibilities for a SOET with a valid 11-skip are shown in fig. 10. Note that these possibilities have the same 'local' structure, the only difference is how the rest of the SOET U is connected to the valid 11-skip. We first show the procedure that should be applied if the 11-skip is of the form in fig. 10a. shown. Also shown are sub-trail of a SOET U labeled by the maximal sub-words that describe them. These sub-trails always begin and end at a vertex in V (R) but their path inside triangle subgraphs is not shown explicitly. Dashed lines are used to indicate that the sub-trails also traverse unspecified further parts of the graph Λ(R). Figures 10a and 10b show the two valid 11-skips. The sub-word X 1 describes some sub-trail of U connecting T v and T s1 . With a slight abuse of notation we denote the endpoints of this trail by v (X) and s (X) 1 . Similarly for X 2 , X 1 and X 2 .
The SOET U in fig. 10a is of the form where where X 1 is the word associated to the sub-trail connecting v (X) and s (X) 1 as seen in fig. 10a and similarly for X 1 , X 2 , X 2 Since our goal is to make this a HAMSOET we need to have that pairs of vertices in V (R) are only consecutive w.r.t. U if they are adjacent in R. This can be done by applyingτ -operations to U at v (s1) and s (u) 1 . The Eulerian tour U after these operations will be described by where the over-set tilde indicates the mirror-inverting of a sub-word. Note that neither (u, v) or (s 1 , s 1 ) are consecutive anymore, but instead (u, s 1 ) and (v, s 1 ) are now consecutive w.r.t. U . To make sure that this procedure works we need to check two things: (1) U is a SOET and (2) and similarly for U we have It is therefore clear that the Eulerian tour U is a SOET. Furthermore the only consecutive pairs of vertices in U which where not consecutive in U are (u, s 1 ) and (v, s 1 ). Since (u, s 1 ) and (v, s 1 ) are edges of R we see that we can iteratively apply this procedure to any valid 11-skip as in fig. 10a and turn the SOET into a HAMSOET. Similarly the SOET in fig. 10b can be turned into a HAMSOET by applying τ -operations to the vertices s (u) 1 and v (s 1 ) . One can explicitly check this by applying the operations to U in fig. 10b which is given by where with everything defined similarly to the case of fig. 10b. Going through a similar argument as above we can show that we can also turn the SOET U into a HAMSOET. This completes the lemma.
Lemma 3.6. Let R be a 3-regular graph and Λ(R) be its triangular-expansion. Also let u, v be adjacent vertices on R.
Let U be a SOET on Λ(R) with respect to V (R). Let X be a maximal sub-word of m(U ) not associated to u and/or v containing u (v) v (u) and describing a sub-trail that makes true skips at T u and T v . Then u and v are consecutive in U and moreover m(U ) contains a sub-word of the form Proof. The situation described in the lemma is described graphically in fig. 11. Because U is a SOET the sub-word v (u) u (v) must be contained exactly twice in m(U ). Note that at most one of these instances can be contained in the maximal sub-word X. The other instance must be contained in a different maximal sub-word Z. This maximal sub-word will be associated to two vertices w 1 , w 2 . Note that since v (u) u (v) ∈ Z and v (u) u (v) ∈ X either we must have that w 1 = u or that the sub-trail described by the maximal sub-word Z makes a true skip at T u . Since T u already contains a true skip (made by the sub-trail described by X) we must have that w 1 = u. We can make the same argument for the vertex v. This means the maximal sub-word Z must be associated to u and v and hence that u, v must be consecutive in U and moreover we have that Figure 11: This figure is a graphical aid for lemma 3.6. Shown are two triangle subgraphs T u , T v (gray triangles) with outer vertices (circles) and the vertices u and u (diamonds) shown. The lemma starts from assuming the existence of the sub-trail described by the maximal subword X, labeled as such in the figure. From this starting point the existence of the maximal sub-word Z associated to the vertices u and v is derived.
Lemma 3.7. Let R be a 3-regular graph and Λ(R) be its triangular-expansion. Also let u, v be adjacent vertices on R.
Also take x 1 , x 2 to be the vertices adjacent to u in R such that x 1 = v, x 2 = v. Let U be a SOET on Λ(R) with respect to V (R). Let Y be a maximal sub-word of m(U ) not associated to u and/or v containing u (x1) x (u) 1 and u (x2) x (u) 2 and describing a sub-trail making a true skip at T u . Also let X be a maximal sub-word associated to u and a vertex x 3 = v that describes a sub-trail making a true skip at v. Then u, v are consecutive and moreover m(U ) contains a sub-word of the form Proof. The situation described in the lemma is described graphically in fig. 12. Because U is a SOET the sub-word v (u) u (v) must be contained exactly twice in m(U ). Note that at most one of these instances can be contained in the maximal sub-word Y and none can be contained in the maximal sub-word X. This means there must be a maximal sub-word Z of m(U ) (different from X and X) containing v (u) u (v) . This maximal sub-word must again be associated with two vertices x,x. If these vertices are not u, v then the sub-trail described by Z must make true skips at T u , T v or both. Since both of these triangle subgraphs already contain true skips this is not possible and hence Z must be associated to u and v which means they are consecutive. Moreover, by construction of Z we have u Y X Figure 12: This figure is a graphical aid for lemma 3.7. Shown are two triangle subgraphs T u , T v (gray triangles) with outer vertices (circles) and the vertices u and v (diamonds) shown. The lemma starts from assuming the existence of the sub-trails described by the maximal subwords Y and X, labeled as such in the figure. From this starting point the existence of the maximal sub-word Z associated to the vertices u and v is derived.

Algorithms
This section is concerned with providing algorithms for various versions of the decision problems STARVERTEXMINOR with input (G, V ) and VERTEXMINOR with input (G, G ). We begin by describing an efficient algorithm for STARVERTEXMINOR whenever the input graph G is distance-hereditary. We prove that this algorithm always terminates and gives correct results. We also analyze its runtime and show that it is O(|V ||V (G)| 3 ). Next we describe an algorithm for STARVERTEXMINOR whenever the input graph G is a circle graph. We prove that this algorithm is fixed-parameter tractable in the size of the input vertex-set V . Finally we prove for the decision problem VERTEXMINOR with arbitrary connected input graphs G and G , that whenever |V (G )| ≤ 3 and V (G ) ⊂ V (G) we have that G < G and provide an efficient algorithm for finding the sequence of local complementations and vertex-deletions that takes G to G .

Star graph as vertex-minor of a distance-hereditary graph
In this section we present an efficient algorithm for deciding whether a star graph on a given set of vertices V is a vertex-minor of a given distance-hereditary graph G. Throughout this section we assume that the graph G is distancehereditary and that V is a subset of its vertices. The algorithm presented in this section will return a sequence of vertices v in V (G), such that τ v (G)[V ] = S V if such a sequence exists and raise an error-flag otherwise, indicating that S V is not a vertex-minor of G. We first present the algorithm in section 4.1.1, analyze its runtime in section 4.1.2 and prove that it is correct in section 4.1.3. An implementation in SAGE [47] of the algorithm can be found at [1].

The algorithm
We first give a rough sketch of the idea behind the algorithm. Remember that the task of the algorithm is to find a sequence of local complementations τ v such that the induced subgraph of τ v (G) on the vertices V is a star graph.
The algorithm starts by choosing a one vertex c in V which will become the center of the star graph on V . It then proceeds by different picking vertices v ∈ V and making them adjacent to c by performing local complementations. After every vertex that is made adjacent the algorithm will check if the induced subgraph on the neighborhood of c is a star graph. If it is not it will attempt to turn it into a star graph by local complementations. If it fails at doing so it will raise an error and if it succeeds it will pick another vertex in V and repeat the procedure until all vertices in V are in the neighborhood of c. We will often call this process of making a vertex v adjacent to c 'adding' the vertex v to the star graph. To understand when the algorithm might fail we now zoom in on the situation where all but one vertex of V has been added to the neighborhood of c. 1. Make f and c adjacent, without changing any edges in G[V \ {f }]. The star graph S V is then a subgraph of the graph, but not necessarily an induced subgraph, since f could be also be adjacent to other vertices in V than c. We will call these edges between f and vertices in V \ {f } bad edges. This first step is the task of algorithm 2 below. Interestingly, this step always succeeds if the graph is connected, even if the graph is not distance-hereditary.
2. Remove the bad edges, without changing any other edges between vertices in V . The removal of the bad edges is the task of algorithm 1 below. Algorithm 1 tries to remove the bad edges by checking a few cases. Thus, one of the main results of this section is to prove that these cases provide a necessary condition for S V being a vertex-minor of G.
We will now describe the two main steps above of the algorithm in more detail. Let's denote the vertices as above and furthermore the current leaves in G[V \ {f }] as V \ {c, f } = {l 1 , . . . , l k }.

Details of step 1:
The vertices c and f are made adjacent by performing local complementations along the shortest path P from c and f , see fig. 13a. The operations along the path P will in fact be either pivots, i.e. ρ (u,v)   Next, we must remove the bad edges in order to turn G[V ] into a star graph. Let G now be the graph with f and c adjacent but with possibly some bad edges.

Details of step 2:
In this step we will remove the bad edges, if we can. A situation where bad edges can be removed, as we will show, is when there exists a vertex u / ∈ V , which is adjacent to all vertices in B but not to any vertex in L. The existence of such a vertex u is thus a sufficient condition for the removal of bad edges. When G is a distance hereditary graph, it turns out that this condition is also necessary, that is if no such vertex u exists, then the star graph on V is not a vertex-minor of G, and we can stop the algorithm. This is shown in detail in section 4.1.3. For this statement to hold L cannot be empty, but this can always be achieved by performing a local complementation at c first if needed, which is done in line 14 in algorithm 1. Assume that there indeed exist such a vertex u, i.e.  Figure 14b shows the graph after performing a local complementation on u which produces a new leaf (white square) and makes the bad edges form a complete graph. This complete graph of bad edges can then be removed by finding a vertex u that is adjacent to all vertices in B (and to none in L) and performing a local complementation at u .
Performing the same step again, i.e. doing a local complementation at another vertex adjacent to all vertices in B \ {f }, will remove all bad edges. We have then produced the star graph on V in two steps.
• Case 2: u and c are adjacent: In this case, if a local complementation is performed at u, some edges between c and vertices in L will be removed, which is not desired. We can solve this by finding another vertex h adjacent to both u and c but not to any other vertex in V , by which we can remove the edge (u, c), see fig. 15. In the following section we show that if there is no vertex h of this form, the star graph is not a vertex-minor of G and we can stop the algorithm.
To prove that the algorithm is correct we need to show that cases checked by algorithm 2 to remove the bad edges actually provides a necessary condition for S V being a vertex-minor of G. To be precise, we will show that a necessary 8 condition for the star graph on V being a vertex-minor of G is where V = B ∪ L ∪ {c} and L is assumed to be nonempty. It is important to note here that this condition is only valid if the graph is in the correct form, i.e. the induced subgraph on V form a star-star graph or a complete-star graph. We formally state that eq. (105) is a necessary condition for the star graph on V being a vertex-minor of G in theorem 4.1, which we prove in section 4.1.3. The theorem uses the notion of star-star and complete-star graphs which we formally define as: Candidates for the u in eq. (105) 24: if U = ∅ then 25: Raise ERROR(S V is not a vertex-minor of G) Actually not needed, only for clarity 26: end if

Runtime of the algorithm
The algorithm described in the previous section checks if a star graph with vertex set V is a vertex-minor of a distancehereditary graph G. Here we show that the runtime of this algorithm is O(|V ||V (G)| 3 ). We will represent subsets of a base-set as unsorted binary lists 9 , where 1 indicates that an element in the base-set is in the represented set and 0 that an element in the base-set is not in the represented set. This will be the case both for sets of vertices and sets of edges. The base-set for sets of vertices will be the set of vertices V (G) of the input graph G and the base-set for edges-sets will be V (G) × V (G). Thus, we assume that the input graph G is given as an unsorted binary list, of length |V (G)| 2 , indicating which edges are in E(G). This allows us to check if an edge (u, v) is in the graph or not in constant time. Furthermore, we assume that the input-set V is also represented as an unsorted binary list, of length |V (G)|, indicating which of the vertices of G are in V . We also assume that the size of |V | is given together with its representation, which allows us to faster create representations of subsets of V .
Sets used internally by the algorithm (B, L and U ) will also be represented as unsorted binary lists together with the size of the sets. The sizes of the sets will be updated accordingly whenever an element is added. Note that B and L are subsets of V and will therefore be represented as unsorted binary lists, of length |V |, indicating which elements of V are in these sets. However, U is not a subsets of V and will therefore be represented by an unsorted binary list of length |V |. Thus, given a vertex v, checking if v is in a set of vertices V can be done in constant time and adding a vertex to a set can be done in constant time (flipping the bit at the corresponding position). Furthermore, iterating over elements in a set can be done in linear time with respect to the base-set, i.e. O(|V (G)|) for V and U and O(|V |) for B and L.
As described, the full algorithm starts by calling algorithm 1, which in turn calls algorithm 2, which again calls algorithm 1 and so on. We will see that the computation that dominates the runtime is updating the graph τ v (G) whenever v is concatenated, as in line 13 of algorithm 1 and line 6 of algorithm 2. We will assume that both algorithm 1 and algorithm 2 have access to a common graph which they can update to τ v (G), whenever v is concatenated, to prevent this from being done for the whole sequence Thus, the total runtime of algorithm 1, except for the recursive call to algorithm 2 in line 6, is O(|V (G)| 2 ) (from line 7 in the for -loop).
To further substantiate the efficiency of the algorithm we give actual run-times for an implementation of the algorithm in fig. 1.

Proof that the algorithm is correct
In this section we prove that the algorithm presented in the previous section works, i.e. it gives a sequence of local complementations v such that τ v (G)[V ] = S V , given a distance-hereditary graph G, if such a sequence exists. It is relatively easy to show that the algorithm gives the desired results when it does not return an error, which we show in section 4.1.3.1. The hard part is to prove that, when the algorithm gives an error-flag it is in fact not possible to produce the star graph, i.e. the star graph is not a vertex-minor of G, which is done in section 4.1.3.2. The notation will be the same as in the previous section, c is a vertex in V and is adjacent to the rest of the vertices in V . The vertices in G[V \ {c}] with degree greater than 0, i.e. the vertices incident on some bad edge, are denoted as the set B.

Algorithm succeeds
In this section we show that if algorithm 1 returns a sequence v, i.e. does not give an error-flag, then τ v (G)[V ] = S V . We start by showing that algorithm 2 always succeeds and gives the desired result, assuming that algorithm 1 works. After performing a pivot ρ (v,u) , i.e. τ u • τ v • τ u , any neighbor of v will become a neighbor of u, except u itself. This means that after the first pivot in line 9 in algorithm 2, i.e. ρ (p1,f ) , f and p 2 will be adjacent. We want to inductively show that this implies that after performing pivots along the whole path, f and c are adjacent. To do this we only need to make sure that a pivot does not remove edges in the later part of the path. More precisely, the pivot ρ (pi,f ) should not remove an edge (p j , p j+1 ) for j > i. The fact that later edges in the path are not removed follows from the properties of the pivot and that the path is a shortest path. Apart from edges incident on u or v, a pivot ρ (v,u) can only flip edges in the set N v × N u . This shows that the pivot ρ (pi,f ) cannot remove an edge (p j , p j+1 ) since neither p j or p j+1 is equal to f or p i or is adjacent to f . If p j or p j+1 would be adjacent to f , then this would not be a shortest path. We also need to make sure that we do not remove the edges from E(S V \{f } ) = {(c, v) : v ∈ V \ {c, f }}, when doing pivots along the path. By the same argument above we have that the pivot ρ (pi,f ) can only remove an edge in E(S V \{f } ) if f is adjacent to a vertex in V \ {c}. This is the reason for the if-statement in line 5 in algorithm 2, where we then just perform a local complementation on the corresponding vertex in v ∈ V \ {c} which will make f and c adjacent. Performing the local complementation on such a vertex v will not remove edges in E(S V \{f } ), since v is a leaf in the induced subgraph on V . Note that there are only two cases where bad edges are created. If f and c are made adjacent by a local complementation on a vertex v ∈ V \ {c}, as in line 7, the bad edge (f, v) will be created. On the other hand, if this is not the case but the last vertex p k is adjacent to some vertices U ⊆ (V \ {c, f }), then the bad edges {(f, u)} u∈U will be created. In both of these cases τ v (G)[V ] will be a star-star graph, see definition 4.1. Note that f can also be adjacent to some vertices in V \ {f }, even before we perform the local complementations, but these edges will still form a star graph with f as the center. If one wants to minimize the number of local complementations and use local complementation instead of pivots along the path, this is in fact possible. The only place where a pivot is needed instead of a local complementation is towards the end of the path, when p i is adjacent to a vertex in V \ {c, f } not on the path.
What is left to show is that if algorithm 1 succeeds and returns a v, then τ v (G) = S V . This is easy to see, since if we perform local complementations on such vertices we are looking for, i.e. u and possibly h in eq. (105), we will remove the bad edges and produce the star graph. If |B| > 2 this has to be done twice, as captured by the loop over i in algorithm 1. The reason for this is that, when doing a local complementation on such a u we complement the induced subgraph G[B]. Since G[B] is a star graph, the induced subgraph after the local complementation will be a complete graph plus a single disconnected vertex which was the center of G[B]. Performing the step once more will then complement the complete graph, without the disconnected vertex, and all bad edges are thus removed.
Note that we have nowhere in this section used the assumption that the graph is distance-hereditary. This implies that if the algorithm succeeds we know that τ v (G) = S V , independently of whether G is distance-hereditary, in fact even independently of the rank-width of G. Furthermore, since algorithm 2 always succeeds to make τ v (G)[V ] connected and from the fact that any connected graph on two or three vertices is either a star graph or a complete graph, this implies that a star graph on any subset of size two or three is a vertex-minor of G, if the vertices are connected in G, which we make use of in section 5. On the other hand, if the algorithm stops and gives an error-flag, then we do not know in general if S V is a vertex-minor of G or not. In the next section we show that if the graph is distance-hereditary and the algorithm gives an error-flag we actually do know that S V is not a vertex-minor of G.

Algorithm gives error
In this section we prove that if algorithm 1 gives an error-flag, i.e. if P(B, L, c) in eq. (105) is false, then the star graph is not a vertex-minor of the input graph. At the steps in the algorithm where the error-flag can be raised, we know that the induced subgraph on V is either a star-star graph (definition 4.1) or a complete-star (definition 4.2) graph as shown in section 4.1.3.1. The proof will follow the following sequence of steps. 4.1.3.3 Proof for a star-star (complete-star) graph of size 4 We will first show in theorem 4.2 that if P(B, L, c) is false and |V | = 4, then S V is not a vertex-minor of G. This will then allows us to prove the statement for the cases where |V | ≥ 4.
Theorem 4.2. Let's assume that G is a distance-hereditary graph with the following induced subgraph (which is both a star-star and a complete-star-graph) (110) There are only two ways for P(B, L, c) to be false; either ( We will consider these two cases separately and prove that if either is true, then T (G) \ V = ∅. In most cases below we will do this by showing that one of the four vertices in V is not in T (G) which shows that T (G) \ V = ∅ since T (G) ≥ 4 by theorem 2.8.
• Assume that |(N 3 ∪ N 4 ) \ {3, 4}| > 1. Since, by assumption N 3 ∩ N 4 = {2}, 3 and 4 does not form a twin-pair. Also, neither 3 nor 4 is not a twin, since the twin-partner would have to be a common neighbor of 3 and 4. Furthermore, neither 3 or 4 is a leaf. Thus, the only way for 3 ∈ T (G), is if 3 is an axil, requiring some vertex not in V being a leaf. 12 Finally, if 3 / ∈ T (G), then there exist a vertex in T (G) which is not in V , since by theorem 2.8 we know that |T (G)| ≥ 4.
-Assume that |N 1 | > 1. Then 1 is not a leaf and 2 is not an axil. Furthermore, since nothing else is connected to 3 and 4, 2 is not a twin. Thus, the only way for 2 ∈ T (G), is if 2 is an axil, requiring some vertex not in V being a leaf. Finally, if 2 / ∈ T (G), then since |T (G)| ≥ 4 by theorem 2.8, there must exist a vertex in T (G) which is not in V .
-Assume that |N 1 | = 1. Then 2 is necessarily a cut-vertex and G \ 2 will contain a connected component with no vertices in V , since |V (G)| > 4,. Thus, there exist a vertex in T (G) which is not in V by corollary 2.8.1.

Case 2:
To prove that if eq. (112) is true then there exists a removable leaf or twin, we consider the following cases. 11 As in section section 2.5, removable means a vertex not belonging to the target vertices V 12 The same for 4.
• Assume that |N 1 | > 1. Then if 2 is an axil, the corresponding leaf cannot be in V , since 1 is not a leaf. Furthermore, since 2 is not a leaf, if 2 ∈ T (G) then 2 has a twin-partner not in V , which is then also in T (G). On the other hand if 2 / ∈ T (G), then there exist a vertex in T (G) which is not in V , by theorem 2.8.
In this case, 3 and 4 does not form a twin-pair. Furthermore, neither 3 or 4 is a leaf. Thus the only only way for 3(or 4) ∈ T (G), is if 3(4) is an axil or a twin, requiring some vertex not in V being a leaf or a twin. Finally if 3(4) / ∈ T (G), then there exist a vertex in T (G) which is not in V , since by theorem 2.8 we know that |T (G)| ≥ 4.
We will for this case show that |T (G) \ V | > 0 by assuming that T (G) = V and arriving at a contradiction. Since this implies that T (G) = V and from the fact that |T (G)| ≥ 4, we know that |T (G) \ V | > 0. Consider the induced subgraph G \ 4 which is also distance-hereditary. 13 From theorem 2.8 we know that Note that by the assumption from eq. (112), any neighbor of 4, except 2, is also a neighbor of both 2 and 3. The removal of 4 cannot therefore create a new leaf in V \ V . The only option left is if there are two vertices v, v ∈ V (G) \ {4}, such that v, v form a twin-pair in G \ 4 but not in G. If v and v are such vertices, it must be the case that 4 is adjacent to exactly one of v and v . Assume without loss of generality that 4 is adjacent to v but not to v. The neighborhoods of these vertices are then Note that the vertices adjacent to 4 are (N 3 ∩ N 4 ) ∪ {3}. Firstly, v cannot be in N 3 ∩ N 4 , since v is then necessarily adjacent to 3 but not to 4 which contradicts the assumption that ( Secondly, v cannot be 3, since v is then necessarily a neighbor of 2 and all vertices in N 3 ∩N 4 , contradicting the second part of eq. (112).

Proof for star-star graphs
We are now able to prove the same statement as in theorem 4.2 but for |V | ≥ 4. The case when G[V ] is a star-star graph is given in theorem 4.3.
For each l ∈ L , let u l be a vertex in V \ {l, c , b 1 , b 2 } such that Q(u l , {b 1 , b 2 }, {l}, c ) is true. We will now show that for at least one of these u l , u l ∈ V \ V and Q(u l , B , L , c ) is true. To show that for at least one u l , u l ∈ V \ V and Q(u l , B , L , c ) is true we will go trough the following steps: 1. Show that ∀l ∈ L : (u l / ∈ V ) (117) 13 Induced subgraphs of a distance-hereditary graph are distance-hereditary. 14 Remember that L = ∅, by definition of a star-star graph.

Show that
3. Show that ∃l ∈ L : (L ∩ N u l = ∅) 4. Fix l ∈ L to be such that the corresponding expression in eq. (119) is true.
If all the statements in the above steps are shown to be true we know that there exist a u l in V \ V such that Q(u l , B , L , c ) and therefore P(B , L , c ) is true. It is important to note that even if we consider the statements in the above steps separately we know that there exist at least one u l that simultaneously satisfy all. To see this, note that we only use a existential quantifier in step 3 and in step 5 we consider a u l which satisfies the corresponding property in step 3. Let's now consider the steps 1 through 5 one by one. 15 We will in these steps often claim that certain small graphs are not distance-hereditary. Verifying this can be done by hand or using our code supplied at [1].
Step 1: is a star graph. Furthermore, since b and l are not adjacent ∀b ∈ B and ∀l ∈ L , u l is not in L . We therefore know that u l / ∈ V .
Step 2: To see that B ⊆ N u l , assume that first that this is not the case, i.e. ∃b ∈ B :b / ∈ N u l . We will now show that this contradicts the distance-hereditary property. Note thatb is not the center of the star graph G[B], because either b 1 or b 2 is the center, since (b 1 , b 2 ) is an edge in G[B]. Let's assume without loss of generality that b 1 is the center of G [B]. Furthermore, let's consider the cases where u l and c are adjacent or not separately.
• Assume that u l and c are not adjacent and consider the following induced subgraph The graph in eq. (121) is not distance-hereditary, since the distance between for exampleb and u l increase if b 1 is removed. This is therefore a contradiction to the assumption that G is distance-hereditary.
• Assume that u l and c are adjacent. We then know that there exist a vertex h l which is adjacent to u l and c , since Q(u l , {b 1 , b 2 }, {l}, c ) is true. First, let's show that h l andb are not adjacent. In fact we will show that h l is not adjacent to any vertex in B, which will be useful in step 5.
Assume the opposite, i.e. h l is adjacent to some vertexb ∈ B \ {b 1 , b 2 }. We already know that h l is not adjacent to b 1 or b 2 , since Q(u l , {b 1 , b 2 }, {l}, c ) is true. Consider therefore the following induced subgraph which is not distance-hereditary and we therefore know that N h l ∩ B = ∅. 15 Step 4 is trivial.
Consider now on the other hand the following induced subgraph, with the knowledge that h l is not adjacent tob which is also not distance-hereditary.
Since in all cases we arrived at a non-distance-hereditary graph we know that B ⊆ N u l .
Step 3: We show that at least for one of the u l , L ∩ N u l = ∅. We will do this by contradiction, assume therefore that ∀l ∈ L : (L ∩ N u l = ∅). Since Q(u l , {b 1 , b 2 }, {l}, c ) is true, we know that {l} ∩ N u l = ∅ for all l ∈ L . Consider now the graph G[L ∪ {u l } l ]. From theorem 4.4 we know that there exist l 1 , l 2 ∈ L and u l3 , u l4 ∈ V \ V , such that u l3 is adjacent to l 1 but not to l 2 and u l4 is adjacent to l 2 but not to l 1 . 16 But this is in contradiction with that the graph is distance-hereditary. To see this, consider the induced subgraph which is not distance-hereditary, independently if the edges (u l3 , u l4 ), (c , u l3 ) and (c , u l4 ) are individually present or not. Since this contradicts the distance-hereditary property we know that ∃l ∈ L : (L ∩ N u l = ∅).
Step 5: Let's assume that u l is then a vertex such that B ⊆ N u l and L ∩ N u l = ∅. If u l is not adjacent to c , then clearly Q(u l , B , L , c ). On the other hand if u l and c are adjacent we know that there exist a h l in N u l ∩N c \ x∈{l,b1,b2} N x . We thus need to show that h l is not adjacent to any vertex in V , other than c . Firstly, h l cannot be adjacent to a vertex in L , since this would violate the distance-hereditary property. To see this, assume that h l is adjacent tõ l ∈ L and consider the following induced subgraph which is not distance-hereditary. This is a contradiction with the distance-hereditary property and we therefore know that N h l ∩ L = ∅. As we already shown in step 2, h l is also not adjacent to any vertex in B . Thus, h l is not adjacent to any vertex in V = B ∪ L ∪ {c }.
We have therefore shown that eq. (115) is true which implies that eq. (114) is true. Finally, as we described in the beginning of the proof, this implies that if P(B , L , c ) is false then S V ≮ G.
Theorem 4.4. Assume G is a graph on the vertices U ∪ L such that U ∩ L = ∅ and U = ∅. Furthermore, assume that for each l in L, there is at least one vertex in U not adjacent to l and for each u in U , there is at least one vertex in L adjacent to u, i.e. G satisfies the following expression R(U, L) = ∀l ∈ L : ∃u ∈ U : u / ∈ N l ∧ ∀u ∈ U : ∃l ∈ L : l ∈ N u (126) Then there exist two vertices u 1 and u 2 in U and two vertices l 1 and l 2 in L such that u 1 is adjacent to l 1 but not to l 2 and u 2 is adjacent to l 2 but not to l 1 . In other words the induced subgraph is of the following form where the dashed edges are individually either present or not.
Proof. We will first show that |L| ≥ 2 and |U | ≥ 2. Pick an element u 1 ∈ U , which exists since U = ∅, by assumption there is a l 1 ∈ L which is adjacent to u 1 . Furthermore there exist a u 2 ∈ U which is not adjacent to l 1 , thus u 1 = u 2 . Finally, by assumption there is a l 2 ∈ L which is adjacent to u 2 , thus l 1 = l 2 . Note, that this does not yet prove the theorem, since u 1 and l 2 might be adjacent.
We will first prove the theorem for |L| = 2 and then use this to prove the general case.

|L| = 2 :
Let's denote the vertices in L by l 1 and l 2 . We first show by contradiction that both vertices in L must have at least one neighbor in U . Assume that l 1 does not have a neighbor in U , then all vertices in U must be adjacent to l 2 by the second part of eq. (126), but then the first part of eq. (126) is false. Thus l 1 has at least one neighbor in U and by symmetry the same is true for l 2 . Now choose such a neighbor of l 1 in U and denote this u 1 . We now show by contradiction that there exist another vertex u 2 ∈ U which is adjacent to l 2 but not to l 1 . Assume that this is not the case, i.e. all vertices in U \ {u 1 } are adjacent to l 1 or not adjacent to l 2 . If a vertex in U is not adjacent to l 2 then it is necessarily adjacent to l 1 , thus by assumption all vertices in U \ {u 1 } are adjacent to l 1 . This is in contradiction with the first part of eq. (126) and the theorem for |L| = 2 follows.
|L| > 2 : We will show that the following is true: (1) G has an induced subgraph as in eq. (127) or (2) there exist a l ∈ L such that G \ l satisfy R(U, L \ {l}). The theorem then follows since if (1) is true the theorem follows directly and if (2) is true we can make the same argument for G \ l for some l ∈ L and then possibly for (G \ l) \ l etc., which at some point will give the case |L| = 2, which we have proven above. Note that if the graph reached by deleting vertices from G, has the graph in eq. (127) as an induced subgraph, then so does G.
To prove that (1) or (2) is true, we show that if (2) is false then (1) is necessarily true. Therefore, assume now that (2) is false, which means that for every choice of l, G \ l does not satisfy R(U, L \ {l}). The only possibility for this to happen, i.e. the deletion of l makes the graph not satisfy eq. (126), is if the deletion of l makes some u ∈ U not adjacent to any vertex in L. It is easy to see that this can only happen if ∃u ∈ U : (L ∩ N u = {l}). Since this should be true for all l ∈ L, we have that if (2) is false, the following is true, But eq. (128) implies that (1) is true. To see this pick two different vertices l 1 and l 2 in L. From eq. (128) we know that there exist a vertex u 1 ∈ U such that L ∩ N u1 = {l 1 } and similarly a u 2 for l 2 . Note that u 1 = u 2 since N u1 = N u2 . Furthermore, since L ∩ N u1 = {l 1 } and L ∩ N u2 = {l 2 } the induced subgraph G[{u 1 , u 2 , l 1 , l 2 }] is as in eq. (127).

Proof for a complete-star graph
Here we prove that if P(B , L , c ) is false, then S V ≮ G for the case where G[V ] is a complete-star graph. We have the following theorem.
Theorem 4.5. Let's assume that G is a distance-hereditary graph and V is a subset V ⊆ V (G) such that the induced subgraph G[V ] is a complete-star graph KS (B ,L ,c ) . Furthermore assume that P(B , L , c ) is false, then S V ≮ G.
Proof. We will prove this by induction on the size of B . The base-case, |B | = 2, is true due to theorem 4.3, since for |B | = 2, the graph G is also a star-star graph. Let's now assume that theorem 4.5 is true for |B | = k. We will now prove that the theorem is true for |B | = k + 1 by showing that where |B | = k + 1 ≥ 3. Then from the induction hypothesis we know that S V \{b} ≮ G for some b ∈ B and the corollary follows, because if S V \{b} is not a vertex-minor of G then neither is S V , since S V \{b} < S V . To show that eq. (129) is true, we instead show the contrapositive statement, i.e.
We now need to show that for at least one of these u Thus, to show that for at least on u b , u b ∈ V \ V and Q(u b , B , L , c ) is true we will go trough the following steps: 1. Show that there can maximally be oneb ∈ B such that ub ∈ B , i.e.
2. Fixb ∈ B to be such that the corresponding expression in eq. (132) is true. 17 3. Show that there can maximally be oneb ∈ B \ {b} such that B N ub , i.e.
4. Fixb ∈ B to be such that the corresponding expression in eq. (133) is true.
5. Use step 1 to 4 to show that there exist a b ∈ B such that i.e. Q(u b , B , L , c ) is true.
Let us now consider the steps 1 through 5 one by one. 18 We will in these steps often claim that certain small graphs are not distance-hereditary. Verifying this can be done by hand or using our code supplied at [1].
Step 1: Here we show that eq. (132) is true. Firstly, if for all b ∈ B we have that u b / ∈ B , then eq. (132) is clearly true, sincẽ b can then be chosen as any element in B . 19 We now show by contradiction that there cannot exist two different verticesb 1 ,b 2 ∈ B , such that ub 1 ∈ B and ub 2 ∈ B . Thus, let's assume that such verticesb i for i ∈ {1, 2}, does exist. Note that, since Q(ub i , B \ {b i }, L , c ) is true, we know that ub i is adjacent to all vertices in B \ {b i } and therefore ub i =b i . Since ub i =b i , we know that ub 1 = ub 1 . Furthermore, from the fact that ub i ∈ B , we know that ub i is adjacent to c and thus, since Q(ub i , B \ {b i }, L , c ) is true, there exist a vertex h i such that 17 Note that this does not imply that ub ∈ B . 18 Step 2 and 4 are trivial. 19 Remember that |B | ≥ 3.
The vertices h i are necessarily different, since h 1 is adjacent to b 1 = ub 1 but not to b 2 and vice versa for h 2 . Now consider the following induced subgraph where b is a vertex in B \ {b 1 , b 2 }, which exists since |B | ≥ 3 by assumption. The graph in eq. (136) is not distance-hereditary, independently if the edge (h 1 , h 2 ) is present or not. Since this contradicts the fact that G is distance-hereditary, we know that eq. (132) is true.
Step 3: Here we show that eq. (133) is true. Firstly, if for all b ∈ B \ {b} we have that B ⊆ N u b , then eq. (133) is clearly true, sinceb can then be chosen as any element in B \ {b}. 20 We now show by contradiction that there cannot exist two different verticesb 1 ,b 2 ∈ B , such that B N ub 1 and B N ub 2 . Thus, let's assume that such verticesb i for i ∈ {1, 2}, does exist. Let's for the remainder of this step denote ub i as u (i) . From the previous steps we know that Thus, by assumption, we have thatb i / ∈ N u (i) . The vertices u (i) are then necessarily different, i.e. u (1) = u (2) , since for exampleb 1 is a neighbor of u (2) but not of u (1) . We will now show that this contradicts the fact that G is distance-hereditary, by considering the following cases: • Assume that u (1) is not adjacent to u (2) and consider the following induced subgraph where b is a vertex in B \ {b 1 ,b 2 }, which exists since |B | ≥ 3 by assumption. The graph in eq. (137) is not distance-hereditary.
-Assume that neither u (1) or u (2) is adjacent to c and consider the following induced subgraph which is not distance-hereditary.
-Assume that exactly one of u (1) and u (2) is adjacent to c and let's assume without loss of generality that it is u (1) that is adjacent to c . Since Q(u (1) , B \ {b 1 }, L , c ) is true and u (1) is adjacent to c , we know that there exist a vertex h 1 such that Consider now the following induced subgraph where b is a vertex in B \ {b 1 ,b 2 }, which exists since |B | ≥ 3 by assumption. The graph in eq. (140) is not distance-hereditary, independently if the edge (h 1 , u (2) ) is present or not.
-Assume that both u (1) and u (2) is adjacent to c . Since Q(u (i) , B \ {b i }, L , c ) is true and u (i) is adjacent to c , we know that there exist a vertex h i such that * Assume that h 1 = h 2 , which implies that h 1 is not adjacent tob 1 orb 2 and consider the following induced subgraph which is not distance-hereditary.
* Assume that h 1 = h 2 and consider the following induced subgraph where b is a vertex in B \ {b 1 ,b 2 }, which exists since |B | ≥ 3 by assumption. The graph in eq. (143) is not distance-hereditary, independently if the edges are individually present or not. To make this statement more transparent, we also provide the adjacency matrix of the graph in eq. (143). The graph in eq. (143) has adjacency matrix where x 1 , . . . , x 5 ∈ {0, 1}. By explicit computation one can check that for any assignment of the variables x 1 , . . . , x 5 , the graph with adjacency matrix as in eq. (145) is not distance-hereditary.
Since in all cases we arrived at a contradiction of the fact that G is distance-hereditary, we know that eq. (133) is true.
Step 5: Here we show that there exist a b ∈ B such that Note that B ⊆ N u b implies u b / ∈ B , thus we can focus on the first property. We prove the statement by contradiction and assume therefore that there exist no such b, i.e. there exist no b ∈ B such that B ⊆ N u b and for which eq. (146) is true. Let's first introduce the set B of vertices in B which satisfy the first of these properties, i.e.
From the previous steps we know that B is not empty. Furthermore, from our assumption we must have that for all b ∈ B, eq. (146) is false, i.e ∀b ∈ B : (u b , c ) ∈ E(G) ∧ ∀h b : Let b now be a fixed element of B. Since u b is adjacent to c and from the fact that Q(u b , B \ {b}, L , c ) is true, we know that there exist a h b such that Note that, eq. (148) together with eq. (149) implies that h b is adjacent to b but to no other vertex in B . We will now show that this leads to a contradiction by considering a vertex b 1 ∈ B \ {b}, such that u b1 / ∈ B , which we showed exists in step 1. Furthermore, let b 2 a vertex in B \ {b, b 1 }, which is necessarily adjacent to u b1 , since Q(u b1 , B \ {b 1 }, L , c ) is true. Let's consider the following cases: • Assume that u b1 is not adjacent to c . By assumption we then have that B N u b 1 , which implies that u b1 is not adjacent to b 1 . Consider now the following induced subgraph which is not distance-hereditary, independently if the edge (h b , u b1 ) is present or not.
• Assume that u b1 is adjacent to c .
-Assume that B ⊆ N u b 1 . By the same argument as for b and u b , we know that there exist a vertex h b1 which is adjacent to u b1 , c and b 1 but not to any other vertex in B . Consider now the following induced subgraph which is not distance-hereditary, independently if the edge (h b , h b1 ) is present or not.
-Assume that B N u b 1 , which implies that b 1 is not adjacent to u b1 since Q(u b1 , B \ {b 1 }, L , c ) is true. Furthermore, we know that there is a vertex h b1 which is adjacent to u b1 and c and possibly to b 1 but no other vertex in B . Consider now the following induced subgraph which is not distance-hereditary, independently if the edges are individually present or not. To make this statement more transparent, we also provide the adjacency matrix of the graph in eq. (152). The graph in eq. (152) has adjacency matrix where x 1 , . . . , x 5 ∈ F 2 . By explicit computation one can check that for any assignment of the variables x 1 , . . . , x 5 , the graph with adjacency matrix as in eq. (154) is not distance-hereditary.
Since in all cases we arrived at a contradiction of the fact that G is distance-hereditary, we know that there exist a b ∈ B such that u b / ∈ B , B ⊆ N u b and such that eq. (146) is true.
We have therefore shown that eq. (130) is true which implies that eq. (129) is true. Finally, as we described in the beginning of the proof, this implies, by induction, that if P(B , L , c ) is false then S V ≮ G.

Fixed-parameter tractable algorithm for unbounded rank-width
In this section we will show that the star vertex-minor problem (STARVERTEXMINOR) is fixed-parameter tractable for circle graphs, in terms of the size of the considered star graph. More specifically, we will show that there exists an efficient algorithm to decide if S V is a vertex-minor of G, given that G is a circle graph and the subset V ⊆ V (G) is of size k. We will do this by showing that we can map this problem in polynomial time to deciding whether the 4-regular multi-graph that defines G has a SOET on the vertices V . This is done in section 4.2.1. We will then give an algorithm that decides whether a 4-regular multi-graph has a SOET on a subset V of its vertices where V = k is fixed. This is done in section 4.2.2. We begin by formally stating the decisions problems considered.
We first define the problem k−STARVERTEXMINOR. We also define the problem k−SOET.
Problem 4.2 (k−SOET). Let F be a 4-regular multi-graph and let V be a subset of V (F ) with |V | = k. Decide whether F allows for a SOET with respect to V Theorem 4.6. k−STARVERTEXMINOR, restricted to circle graphs, is in P.
Proof. This will follow from theorem 4.7 which provides an efficient mapping of every circle graph instance of k−STARVERTEXMINOR to a corresponding instance of k−SOET. By corollary 4.9.1 k−SOET is in P and hence so is k−STARVERTEXMINOR. An efficient algorithm for k−SOET is given in algorithm 4.
This theorem has the following corollary Corollary 4.6.1. STARVERTEXMINOR is fixed-parameter tractable in the size of the input vertex-set V if the input graph G is a circle graph.
The existence of this fixed-parameter tractable algorithm is theoretically interesting but it is not likely to be of practical use. This is so because while the algorithm is efficient in the size of the input graph G it suffers from a hidden constant that is of size O(k! · (f (k) O(kf (k)) )), where f (k) = 2 2 O(k 2 ) making its practical implementation unlikely.

Mapping k−STARVERTEXMINOR to k−SOET
In this section we will prove that there exists an efficient mapping from instances of k−STARVERTEXMINOR that are circle graphs to k−SOET. This is formalized in the following theorem, the proof of which also provides a prescription of the algorithm that defines the mapping.
Theorem 4.7. Let (G, V ) be an instance of k−STARVERTEXMINOR and let G be a circle graph. These is an efficient mapping from this instance to an instance of k−SOET and moreover the instance (G, V ) is a yes-instance of k−STARVERTEXMINOR if and only if its image under the mapping is a yes-instance of k−SOET.
Proof. We will prove this by providing an explicit mapping. An instance (G, V ) of k−STARVERTEXMINOR, where G is a circle graph and V a vertex set, can be mapped to an instance of k−SOET by the following two steps: • Find a double occurrence word X with letters in V (G) such that G = A(X). This can be done in time O(|V (G)| 2 ) by using Spinrad's algorithm [46].
• Construct a 4-regular multi-graph F , such that X = m(U ) for some Eulerian tour U on F . As shown in [12], this can be done in the following way: -Let C X be a cycle graph with the vertices labeled as the consecutive letters of X.
-Contract every pair of vertices with the same label, while keeping all the edges. Note that this step can create multi-edges or self-loops. This step can be done in time O(|V (G)|) by adding the corresponding rows of the adjacency matrix and deleting one row and one column.
The graph obtained from these steps is then a 4-regular multi-graph F with a Eulerian tour U , such that X = m(U ). Therefore, constructing the 4-regular multi-graph F , given X, can also be done in time O(|V (G)| 2 ).
One can see that the above algorithm runs in O(|V (G)| 2 ) and given a circle graph G outputs a 4-regular graph F that has a Eulerian tour U such that A(U ) = G. From corollary 2.6.1 we then know that S V < G if and only if F allows for a SOET with respect to V . Using the above two steps we see that any circle graph instance of k−STARVERTEXMINOR can be mapped to k−SOET in time O(|V (G)| 2 ).
Given this mapping, the next logical step is to find an efficient algorithm for k−SOET. This is done in the next section.

k−SOET is in P
In this section we will prove that k−SOET is in P. We will do this by explicitly writing down an efficient algorithm. This algorithm will make use of an algorithm which solves a well known graph problem we call k −DPP. k −DPP, for k -Disjoint Path Problem is formally defined as follows. Robertson and Seymour proved that there exist an efficient algorithm for solving k −DPP, if k is fixed. Their proof of correctness is based on 23 papers named Graph minors. I , . . . , Graph minors. XXIII [44]. In [31] an improved algorithm for k −DPP is given, with a much smaller hidden constant describing the scaling in terms of k . This hidden constant is still quite large as running time for the algorithm in [31] is (f (k ) O(k f (k )) )n O(1) , where n is the number of vertices in the graph and f (k ) = 2 2 O(k 2 ) We will discuss an algorithm that solves k−SOET efficiently. It will use an algorithm for k −DPP as a subroutine, calling it a constant number of times.
The algorithm for k−SOET is sketched as follows. Let F be a 4-regular multi-graph and let V ⊂ V (F ) be of size k. Recall that a SOET is a Eulerian path that visits the vertices in V in some order. A necessary condition for a SOET to exist is that there are, for some ordering of the vertices in V two edge-disjoint trails from the first vertex in V to the second vertex in V and from the second to the third and so on. If one is given such a collection of paths it is not hard to see that one can connect these paths to each other to form a tour and moreover extend this total tour to be Eulerian. Hence we could use the k −DPP algorithm described above to to find such a tour by finding all the edge disjoint trails that connect the vertices in V .
There are two problems with this. The first problem is that a SOET with respect to V can exist with respect to any possible ordering of the set V . This means that in order to find this tour we might have to apply k−DPP to all k! different orderings. This is a large overhead but acceptable since we are only looking for an algorithm that is efficient for fixed k. The second problem is that the k −DPP algorithm expects k pairs of vertices and requires these pairs to be different.
This means we cannot input the same vertex-pair twice to get two edge disjoint paths. We can resolve this at some overhead by running the k −DPP algorithm on a modified multi-graph H. This multi-graph, which we call a SOET splitting of F , is created by taking each vertex v i in V and splitting it into two vertices v As an example we could have for instance that e 1 , e 2 are incident on v i . We must consider all possible choices of pairings here (of which there are 6) and since there are k vertices on which to perform this procedure (each vertex in V ) there are 6 k such SOET splittings of F (which are not 4-regular anymore). This means we must call the 2k−DPP subroutine 6 k · k! times to account for all possible orderings of the SOET.
Let's make this a little more rigorous. We begin by defining the notion of a SOET-splitting of a 4-regular multi-graph F with respect to a set V .
We label the two vertices that originate from a vertex v ∈ V as v (a) and v (b) . Note that we have not specified how to connect the edges that were originally incident on v to v (a) and v (b) . There are six possible ways to do this for each v ∈ V and each choice leads to an a priori distinct SOET-splitting multi-graph H.
We also define the subroutine k −DPP

FALSE otherwise
Note that we have only specified the inputs and outputs of this subroutine. For details on the inner workings of this algorithm see [31]. Using this subroutine we can write down an algorithm for k−SOET. We now prove that this algorithm returns TRUE if and only if the multi-graph F allows for a SOET with respect to the vertex-subset V . We begin by proving that if the algorithm returns TRUE the multi-graph F allows for a SOET with respect to the vertex-subset V . We have the following theorem.
Theorem 4.8. Let F be a connected 4-regular multi-graph and let V ⊂ V (F ) be a subset of its vertices with |V | = k. If algorithm 4 returns TRUE then F allows for a SOET with respect to the vertex set V .
Proof. Let F be a connected 4-regular multi-graph and let V ⊂ V (F ) be a subset of its vertices with |V | = k. If algorithm 4 returns true this means there exists a SOET-splitting H of F and a permutation π of the set [k] such that there exist 2k edge-disjoint paths in H that connect the vertices v  π (2) and v (a) π(3) and so forth. Undoing the SOET-splitting operation that defines F we can see that these edge-disjoint paths can be attached to one another to form a closed trail 21 (a tour) U on F that visits all vertices of V twice in the order v π(1) , . . . , v π(k) . This is not yet a Eulerian tour however. To construct a Eulerian tour out of the tour U we consider the multi-graphĤ obtained from F by deleting all vertices in V (looking at the induced subgraph F [V \ V ]) and subsequently removing all edges in the tour U from the remaining multi-graph. The resulting multi-graph will consist of multiple connected components. These connected components will either consist of a single vertex or will be multi-graphs with vertices of degree two and four. This means all these connected components are Eulerian. Moreover each of these connected components will contain a vertex that is also a vertex in the tour U . Consider on each of these connected components then a Eulerian tour. These tours will thus form tours on the original multi-graph F as well and since all such tours have at least one vertex in common with the tour U we can extend U to a Eulerian tour on the multi-graph F by for each connected component cutting U at such a vertex and inserting the tour originating from the connected components of G. This means that U can be turned into a Eulerian tour and hence there exists a SOET on the multi-graph F . This proves the theorem.
Next we prove the converse statement. Theorem 4.9. Let F be a connected 4-regular multi-graph and let V ⊂ V (F ) be a subset of its vertices with |V | = k. If F allows for a SOET with respect to V then algorithm 4 will return TRUE.
Proof. Let U be a SOET on F with respect to V and without loss of generality assume that U traverses the vertices of V in the order v 1 , v 2 , . . . , v k . Hence for the vertices v 1 , v 2 there are 2 sub-trails U 1 ) might not be a path, but one can easily pick a subset of the edges in U (a) 1 21 Note that a path is also a trail.
which gives a path, for example as the shortest path P 1 . We can make the same argument for the vertices v 2 , v 3 and so forth. By the definition of SOET splittings there must thus exist a SOET splitting G of F with respect to V such that the path P 1 )} returns TRUE. Since algorithm 4 runs over all possible orderings of V and all possible SOET splittings this particular call to 2k−DPP will always happen and hence algorithm 4 will return TRUE. This proves the theorem.
This leads to the following corollary. This corollary then leads to theorem 4.6.

Connected vertex-minor on three vertices or less
In this section we show that a fixed connected graph with vertices V is a vertex-minor of a connected graph G if V ⊆ V (G) and |V | ≤ 3. Furthermore we provide an algorithm that reduces the number of operations one need to transform G to the desired graph.
Theorem 5.1. Let G be a connected graph and G be a connected graph with vertices V , such that |V | ≤ 3. Then we have that Proof. Note that if G < G then clearly V ⊆ V (G). Assume therefore that V ⊆ V (G). Let's denote the vertices in V (G)\V as U = (v 1 , v 2 , . . . , v n−k , where n = |V (G)| and k = |G |. We will now show that G < G by finding a sequence of operations P = P v n−k •· · ·•P v1 , where P vi ∈ {X v , Y v , Z v }, such that each intermediate graph G (i) = P vi •· · ·•P v1 (G) is connected. Since any connected graph with vertices V , for |V | ≤ 3, is either a star graph or a complete graph, which are LC-equivalent, this shows that G < G. Let's therefore consider such an intermediate graph G (i) for some i ∈ [n − k] and the next vertex v i+1 . We will now show that G (i) \ v i+1 or τ vi+1 (G (i) ) \ v i+1 is connected. This will be done by considering the following two cases: Assume that G (i) \ v i+1 is not connected: This means that v i+1 is a cut vertex. Let G 1 be a connected component in the graph G (i) \ v i+1 and G 2 the rest of the vertices, see eqs. (157) and (158) for an illustration. Since v i+1 is a cut vertex, we know that no vertex in N vi+1 ∩ V (G 2 ) is adjacent to any vertex in N vi+1 ∩ V (G 1 ), in the graph G (i) . Thus, all vertices in N vi+1 ∩ V (G 2 ) are adjacent to all vertices in N vi+1 ∩ V (G 1 ) in the graph τ vi+1 (Ĝ) \ v i+1 , showing that this graph is connected, see eq. (159).
Assume that τ v (Ĝ) \ v is not connected: This case follows from the previous. To see this, letG be the graph τ v (G (i) ). From the above case we then know that τ v (G) \ v i+1 is connected. But this graph is exactly G (i) \ v i+1 since local complementations are involutions and the theorem follows.
From theorem 5.1 we can easily formulate an algorithm that finds a sequence of operations taking G to a desired connected graph G on V , by simply checking recursively ifĜ \ v or τ v (Ĝ) \ v is connected. However, if the vertices in V are already close in G and G is a very large, it would be practical to not have to consider all the vertices in G.
Next, we present a more practical algorithm to find a sequence of local complementations and vertex-deletions that take some graph G to a star graph 22 on the vertices a, b and c. The algorithm performs the following steps: • Find the shortest path P 1 between a and b and connect these by doing τ -operations along this path. This first step already give an algorithm for creating a star graph on vertices a and b, see algorithm 5.
• Then do the same with b and c by finding the shortest path P 2 between b and c. The question is now whether we removed the edge between a and b while connecting b and c. This could only happen if P 2 goes through a vertex which is a common neighbor of a and b. Furthermore, since P 2 is the shortest path from c to b this could only be the case for the last vertex on the path, before b.
-If c is connected to a before the last local complementation on P 2 , then stop, since the induced graph is already connected.
-Assume that c is not connected to a before the last local complementation is performed. So in this case, after performing the local complementation along P 2 , a and b will not be connected but they will both be connected to c.
The full protocol is formalized in algorithm 6.  5: Find the shortest path P 1 between a and b 6: Perform τ p for all p ∈ P 1 \ {a, b} 7: Find the shortest path P 2 = (p 0 = b, p 1 , . . . , p n , p n+1 = c) 8: for i in 1, . . . , n do 9: if a and c are not adjacent then

Conclusion
We have shown that deciding if a graph state |G can be obtained from another graph state |G using LC+LPM+CC is NP-Complete, by showing that VERTEXMINOR is NP-Complete. The computational complexity of VERTEXMINOR was previously unknown and was posted as an open problem in [18]. It is important to note that our results are for labeled graphs, since vertices correspond to physical qubits in the case of transforming graph states.
We presented an efficient algorithm for STARVERTEXMINOR when the input graph is restricted to be distancehereditary. It would be interesting to know if the same approach generalize to qudit graph states described by weighted graphs of low rank-width.
We presented an efficient algorithm for k−STARVERTEXMINOR on circle graphs, that is the problem of deciding if the star graph on a subset of vertices V , with |V | = k, is a vertex-minor of another graph. However, the computational complexity of k−STARVERTEXMINOR or k−VERTEXMINOR on general graphs is still unknown.
Below we list some more open questions regarding the computational complexity of deciding how graph states can be transformed using local operations: • Is the problem still NP-Complete if one allows for arbitrary local operations and classical communication (LOCC), instead of restricting to only LC + LPM + CC? Is there an efficient algorithm for states with bounded Schmidtrank width also in this case? It has been shown that many graph state are equivalent under LC if and only if they are equivalent under stochastic LOCC (SLOCC) [51]. An interesting question is therefore whether graph states described by distance-hereditary or circle graphs fall into this category.
• What is the computational complexity of deciding if a graph state |G can be transformed in to another |G if the qubits are partitioned into multiple local sets, within which multi-qubit Clifford operations and multi-qubit Pauli measurements can be performed.
• How do the computational complexity results of this paper relate to the complexity of code switching in stabilizer quantum error correction codes, e.g. fault-tolerant interconversion between two stabilizer codes [27]? More precisely, can one re-use the techniques presented here, to prove that it is computationally hard to decide whether one stabilizer code can be fault-tolerantly transformed into another. We conjecture that this could be possible, since many stabilizer codes (such as topological stabilizer codes) have a notion of 'locality' with fault-tolerant conversions being the conversions that respect this locality [37].
While finishing this work we became aware of work by F. Hahn et al [29], where specific instances of VERTEXMINOR are considered in the context of quantum network routing.