The average cut-rank of graphs

The cut-rank of a set $X$ of vertices in a graph $G$ is defined as the rank of the $\lvert X\rvert \times\lvert V(G)\setminus X\rvert$ $0$-$1$ matrix over the binary field whose $(i,j)$-entry is $1$ if and only if the $i$-th vertex in $X$ and the $j$-th vertex in $V(G)\setminus X$ are adjacent. We introduce the graph parameter called the average cut-rank of a graph, defined as the expected value of the cut-rank of a random set of vertices. We prove that graphs of bounded average cut-rank are well-quasi-ordered by the induced subgraph relation. In other words, every infinite sequence of graphs $G_1,G_2,\ldots$ of bounded average cut-rank admits a pair $i<j$ such that $G_i$ is isomorphic to an induced subgraph of $G_j$. As a corollary, we prove that every hereditary class of graphs can be characterized by a finite list of forbidden induced subgraphs if components of graphs in the class have bounded number of maximal cliques or maximal independent sets. Another consequence of the above well-quasi-ordering result is that for each real $\alpha$, the list of induced-subgraph-minimal graphs having average cut-rank larger than (or greater than) $\alpha$ is finite. We further refine this by providing an upper bound on the size of obstruction and a lower bound on the number of obstructions for average cut-rank at most (or smaller than) $\alpha$ for each real $\alpha\ge0$. Finally, we describe explicitly all graphs of average cut-rank at most $3/2$ and determine up to $3/2$ all possible values that can be realized as the average cut-rank of some graph.


Introduction
We assume, in this paper, that all graphs are undirected, finite, and simple, which means every graph has no parallel edges as well as no loops. A well-quasi-ordering is a quasi-order in which there is neither an infinite strictly decreasing sequence nor an infinite set of incomparable elements. A class S is closed by ≤ if x ∈ S and y ≤ x implies y ∈ S.
One of the deepest results in graph theory is the Robertson-Seymour theorem stating that the class of all graphs is well-quasi-ordered by the minor relation [20]. However, a variety of other extensively investigated graph relations such as induced subgraphs and topological minors do not well-quasi-order all graphs. Therefore, a central question on this topic is: Given a relation ≤, which classes of graphs are well-quasi-ordered by ≤? There are two common approaches for this question: For the second approach, Ding [9] proved that a class of graphs of bounded tree-depth is wellquasi-ordered by the induced subgraph. 1 This approach seems more difficult perhaps because of the obvious antichain consisting of all cycles.
The following question is one of our main problems.
Question 1. Can we find a nontrivial graph parameter π such that a class of graphs of bounded π is well-quasi-ordered by the induced subgraph relation?
We introduce the average cut-rank of a graph for this purpose. Before defining the average cut-rank, we review the cut-rank function of a graph, introduced by Oum and Seymour [18]. The cut-rank function of a graph G is a function ρ G : 2 V (G) → Z that maps every subset X of V (G) to the rank of an |X| × |V (G) \ X| 0-1 matrix over the binary field whose rows are indexed by X and columns are indexed by V (G) \ X such that the (i, j)-entry is 1 if and only if the i-th vertex in X is adjacent to the j-th vertex in V (G) \ X. The average cut-rank of a graph G, denoted by Eρ(G), is the expectation of ρ G (X) for a uniformly chosen random subset X ⊆ V (G).
We prove that a class of graphs of bounded average cut-rank is well-quasi-ordered by the induced subgraph relation, providing one interesting answer for Question 1.
Theorem 1.1. For every real α, the class of graphs of average cut-rank at most α is well-quasiordered by the induced subgraph relation.
A particular consequence of this result is that there are only finitely many forbidden induced subgraphs for the class of graphs in which each component has a bounded number of maximal cliques or maximal independent sets.
Note that for a vertex v of G, Eρ(G − v) ≤ Eρ(G) ≤ Eρ(G − v) + 1. Together with this easy fact, Theorem 1.1 implies that for each real α, there are only finitely many induced-subgraph-minimal graphs of average cut-rank at least α up to isomorphism, because those graphs have average cutrank at most α + 1.
We show in Appendix A that for every fixed rational ε ∈ [0, 1), x n (ε) grows as Θ(2 4n 2 +20n ), where the constant factor possibly depends on ε. However, if ε is irrational then the situation seems much more complicated. Now we are ready to present our second theorem.
Theorem 1.2. For every α ≥ 0 and a graph G without isolated vertices of average cut-rank at least α, if |V (G)| ≥ x ⌊α⌋ ({α}), then G has a proper induced subgraph of average cut-rank larger than α. 2 Theorem 1.2 implies that induced-subgraph-minimal graphs of average cut-rank at least α have bounded number of vertices for each α. Our third theorem shows that the number of such graphs cannot be too small. Indeed we prove a stronger statement in terms of vertex-minors. The vertexminor relation (see [16]) is a graph containment relation that is weaker than the induced subgraph relation, in the sense that if H is an induced subgraph of G, then H is a vertex-minor of G. We will review the detailed definition in Section 2. Our third theorem says that if we have a set S of graphs 1 In fact, Ding proved it for the class of graphs with no P k subgraph, but a class C of graphs has bounded tree-depth if and only if all paths of graphs in C have bounded length, see [15,Section 6.2]. Also see [15,Section 6.10]. 2 We remark that this in fact implies that there is no graph G having average cut-rank exactly α and |V (G)| ≥ x ⌊α⌋ ({α}). characterizing average cut-rank at most α in terms of forbidding graphs in S as a vertex-minor, then |S| cannot be too small. Theorem 1.3. There is some universal constant c > 0 so that the following holds. For every ε ∈ [0, 1) and n ≥ 0, let S be a set of graphs such that the average cut-rank of a graph G is at most (or less than) ε + n if and only if no graph in S is isomorphic to a vertex-minor of G. Then S contains at least 2 cn log(n+1) graphs.
Our final theorem characterizes graphs of average cut-rank at most 3/2 completely and determines all possible reals up to 3/2 that can be realized as the average cut-rank of some graph. For two graphs G and H, let G + H be the disjoint union of G and H, and for an integer m, let mG be the disjoint union of m copies of G. For every k ≥ 0, let E k be K 1,k+1 with one edge subdivided. Theorem 1.4. Let G be a graph with no isolated vertices. Then G has average cut-rank at most 3/2 if and only if it is isomorphic to a vertex-minor of one of P 5 , 3K 2 , 2P 3 , K 1,k+1 , K 2 + K 1,k+1 , and E k for k ≥ 0. In addition, the set of all possible values for average cut-rank of graphs in the interval [0, 3/2] is This paper is organized as follows. In Section 2 we recall basic definitions and results. In Section 3 we discuss an equivalence relation involving cut-rank functions. We introduce and prove basic tools on the average cut-rank in Section 4. Sections 5, 6, 7, and 8, respectively, present the proofs of Theorems 1.1, 1.2, 1.3, and 1.4. We conclude the paper with two additional remarks in Section 9.

Basic notions on graphs.
For all positive integers k, let P k be the path on k vertices, C k be the cycle on k vertices, K k be the complete graph on k vertices, and K m,k be the complete bipartite graph on m vertices one side and k vertices the other side. For the star K 1,k , we call the vertex at the singleton side the central vertex. (If k = 1 then we fix one vertex to be called central.) For a graph G, denote V (G), E(G), A(G), respectively, be its vertex set, edge set, and adjacency matrix. For any S, T ⊆ V (G), let N G (S, T ) be the set of vertices in T adjacent to at least one member in S.
In G, a vertex is isolated if it has degree zero, and a leaf if it has degree one.
Let G[S] be the subgraph of G induced on the vertex set S; in this case we say G[S] is an induced subgraph of G, and set G − S := G[V (G) \ S] as well as G − v := G − {v}. For any two disjoint subsets X, Y of G, denote by G[X, Y ] the induced bipartite subgraph of G with bipartition (X, Y ) consisting of edges having one end in X and the other in Y . For simplicity, set |G| := |V (G)|, and we sometimes write A G instead of A(G).
Let the complement of G, denoted by G, be the graph with vertex set V (G) and edge set {uv : If, in addition, they are adjacent then we call them true twins, otherwise we call them false twins.
In G, a subset S ⊆ V (G) is a clique if every two vertices in S are adjacent, and an independent set every two vertices in S are nonadjacent.
For two disjoint subsets A, B ⊆ V (G), A is complete to B if every vertex in A is adjacent to all vertices of B and A is anticomplete to B if every vertex in A is non-adjacent to all vertices of B.
For a subset S of V (G), identifying S is the operation of replacing all vertices in S by a new vertex and joining it to everybody in N G (S, V (G) \ S). For an equivalence relation ≡ on V (G), the quotient graph of G induced by ≡ is the graph obtained from G by identifying each equivalence class C of (V (G), ≡) to a vertex denoted by C.
For two graphs G 1 and G 2 , we say G 1 is isomorphic to G 2 and write is an edge of G 2 if and only if uv is an edge of G 1 .

2.2.
Local complementations, vertex-minors, and pivot-minors. For a graph G and its vertex v, let G * v be the graph obtained from G by switching all adjacencies between neighbors of v. To be precise, two vertices x and y are adjacent in G * v if and only if in G, either (1) they are adjacent and at least one of them is not adjacent to v, or (2) they are non-adjacent but both are adjacent to v.
. We call such an operation the local complementation at v. We say two graphs are locally equivalent if one can be obtained from the other by a series of local complementations.
We say that a graph H is a vertex-minor of G if it can be obtained from G by a series of local complementations and vertex deletions. A simple observation points out that given such a series, we may rearrange the operations so that all the local complementations are executed before the vertex deletions without changing the output graph. Thus, if H is a vertex-minor of G, then H is actually an induced subgraph of a graph locally equivalent to G.
For an edge uv of G, the pivot of G on uv, denoted by G ∧ uv, is an operation to obtain a graph from G by three local complementations, G * u * v * u. This is well defined because 2.3. Cut-rank. For a matrix M := (m ij : i ∈ R, j ∈ C), let rank(M ) be its rank and nullity(M ) be its nullity. If X ⊆ R and Y ⊆ C, denote by M [X, Y ] the submatrix of M obtained by taking the rows indexed by X and the columns indexed by For a graph G and two disjoint subsets X and Y , let us write ρ * G (X, Y ) = rank(A G [X, Y ]) where A G is considered as a matrix over the binary field. The cut-rank function of a graph G is a function ρ G : In this paper we need the following property of cut-rank functions, which shows that local complementations preserve the cut-rank function of a graph G. Proposition 2.1 (Oum [16,Proposition 2.6]). For a graph G and v ∈ V (G), we have ρ G (S) = ρ G * v (S) for all S ⊆ V (G).

2.4.
Well-quasi-ordering, forbidden lists, and labeled induced subgraphs. Given a set X and a relation ≤ on X , (X , ≤) is a quasi-order if (i) for every x ∈ X we have x ≤ x; (ii) for any x, y, z ∈ X , if x ≤ y and y ≤ z then x ≤ y ≤ z. We say two elements x, y of X are comparable if x ≤ y or y ≤ x. We say ≤ is a well-quasi-ordering on X , or X is well-quasi-ordered by ≤, or (X , ≤) is a wellquasi-order, if for every infinite sequence {x n } n≥0 of elements of X , there are indices i < j satisfying An antichain is a subset of X having no two comparable elements. A subclass S of X is closed by ≤ if y ∈ S and x ≤ y imply x ∈ S. An antichain C is called a forbidden list for S by ≤ if for all x ∈ X , x belongs to S if and only if there is no y ∈ C satisfying y ≤ x. When X is a class of graphs, X is hereditary if X is closed under induced subgraphs; that is, if G ∈ X and H is an induced subgraph of G then H ∈ X .
Let (Q, ≤) be a well-quasi-order. A Q-labeled graph, or simply a labeled graph (G, l G ) is a graph G equipped with a label l G (v) from Q for each vertex v of G. A labeled graph (H, l H ) is a labeled induced subgraph of (G, l G ) if H is an induced subgraph of G and l H (v) ≤ l G (v) for all v ∈ V (H). We say a class X of graphs is well-quasi-ordered by labeled induced subgraphs if for all well-quasiorder (Q, ≤), the class of labeled graphs (G, l G ) for G ∈ X is well-quasi-ordered by the labeled induced subgraph relation.
The following lemma is called Higman's lemma. . Let ≤ be a well-quasi-ordering on X . Let X <∞ be the set of all finite sequences of elements in X . For any Here is a very useful and well-known characterization of a well-quasi-ordering. (i) A set X is well-quasi-ordered by ≤.
(ii) Every infinite sequence a 1 , a 2 , . . . of elements in X has i < j such that a i ≤ a j .
(iii) Every infinite sequence in X has an infinite increasing subsequence.
The following corollary is an easy consequence of Proposition 2.3, also implied by Proposition 2.2.

An equivalence relation involving cut-rank functions
An attached star in a graph G is an induced subgraph isomorphic to a star whose noncentral vertices are leaves in G. In other words, an attached star in G is an induced subgraph, say G[S], isomorphic to a star such that the set of noncentral vertices is anticomplete to V (G) \ S. The size of an attached star is the number of its vertices.
In G, let ≡ G be a binary relation on V (G) such that for x, y ∈ V (G), It is easy to see that x ≡ G y if and only if one of the following holds: (i) x and y are twins in G, or (ii) one of them is a leaf in G whose unique neighbor is the other. Furthermore, ≡ G is in fact an equivalence relation on V (G), as shown by the following.
Proposition 3.1. The relation ≡ G is an equivalence relation on V (G). Moreover, each equivalence class of (V (G), ≡ G ) is one of the following types in G: the vertex set of an attached star, a clique of true twins, and an independent set of false twins.
Proof. By definition, it is obvious that for x, y ∈ V (G), x ≡ G x and if x ≡ G y then y ≡ G x. Thus ≡ G is reflexive. Now, let x, y, z be distinct vertices in G satisfying x ≡ G y and y ≡ G z. There are three cases to consider.
(1) d G (y) = 0. We have x, y, z are isolated in G and x ≡ G z.
(2) d G (y) = 1. If N G (y) ⊆ {x, z} then trivially x, y, z are leaves in G with a unique common neighbor, so x ≡ G z. If N G (y) ⊆ {x, z}, we may assume that xy ∈ E(G) and yz / ∈ E(G), then y, z are twins in G which implies that z is a leaf in G whose unique neighbor is x, and thus x ≡ G z.
(3) d G (y) ≥ 2. Now, if one of x, z is a leaf in G then so is the other and we deduce x ≡ G z. If both of x, z are leaves in G then {x, y} and {y, z} are two pairs of twins in G which implies easily that x, z are also twins in G, so x ≡ G z.
Hence ≡ G is transitive, so it is an equivalence relation on V (G), as desired. Now let C be an equivalence class in (V (G), ≡ G ). If there is a vertex x in C which is a leaf in G then its unique neighbor, say y, must also be in C; so every vertex in C \ {x, y}, being a twin of x, is also a leaf in G whose unique neighbor is y, which implies that G[C] is an attached star in G.
On the other hand, if C contains no leaves in G, then necessarily they are pairwise twins, but now it is easy to show that C is either a clique of true twins or an independent set of false twins in G. This completes the proof.
In addition, an immediate consequence of Proposition 2.1 is that local complementations preserve every equivalence class in (V (G), ≡ G ).
In other words, Eρ(G) is the expected value of ρ G (S) where S is chosen uniformly at random among all subsets of V (G). Note that due to the symmetry of ρ G , Eρ(G) is a rational number whose denominator in closed form is a positive integer dividing 2 |G|−1 .

4.2.
Basic tools. This subsection deals with basic tools that will be needed in the next sections.
The first proposition deals with the average cut-rank of complete graphs and complete bipartite graphs, including the stars K 1,k .
and is a matrix with all entries 1 otherwise. Therefore thus ρ G (S) = 2. By the same argument, we see that in the other cases, except the cases (S A , S B ) = (A, B) and (S A , S B ) = (∅, ∅) in which ρ G (S) = 0, we always have ρ G (S) = 1. Therefore Now we let m = 1 to get Eρ(K 1,k ) = 1 − 2 −k = Eρ(K k+1 ).
We prove the following result, which shows that the average cut-rank function is preserved under taking local complementations and strictly increasing with respect to the vertex-minor relation among graphs with no isolated vertices.  Proof. Since H is a vertex-minor of G, H is an induced subgraph of G ′ locally equivalent to G. By Proposition 2.1, Eρ(G) = Eρ(G ′ ). Because a vertex remains isolated after each local complementation, it suffices to consider when H is an induced subgraph of G. Let U := V (G) \ V (H). Let S be a subset of V (G) chosen uniformly at random. So S \ U is a random subset of V (H). We have If all the vertices in U are isolated in G then they contribute nothing to every ρ G (S) and hence ρ G (S) = ρ H (S \ U ) for all S ⊆ V .
If there is some vertex in U , say v, having at least one neighbor in V (H), then any subset S of U containing v or any subset S of V (G) containing V (H) but v satisfies ρ G (S) ≥ 1 while ρ H (S \ U ) = 0. On the other hand, if every vertex in U has no neighbor in V (H), then G[U ] has at least one edge, say uv for u, v ∈ U . Then any subset S of V (G) such that S contains only one of which completes the proof.
The following result shows that 1 − 2 1−|G| is in fact the smallest possible average cut-rank of any graph G with no isolated vertices. The equality holds if G is a complete graph or a star, by Proposition 4.1.

Proposition 4.3.
A graph G without isolated vertices has average cut-rank at least 1 − 2 1−|G| . The equality holds if and only if G is a star or a complete graph.
Proof. If G is connected, then for every nonempty proper subset S of V (G), G[S, V (G) \ S] has at least one edge, hence ρ G (S) ≥ 1. Because there are 2 |G| − 2 subsets S of this type, we obtain If G is disconnected, then since G has no isolated vertices, G contains an induced subgraph isomorphic to 2K 2 . It follows that, by Proposition 4.2, Now we consider the equality case. The preceding argument shows that if Eρ(G) = 1 − 2 1−|G| , then G is necessarily connected and ρ G (S) = 1 for all nonempty proper subsets S of V (G). In particular, it follows that for all x, y ∈ V (G), we have ρ G ({x}) = ρ G ({y}) = ρ G ({x, y}) = 1, or equivalently x ≡ G y. Therefore, (V (G), ≡ G ) has only one equivalence class, so Proposition 3.1 implies that G is a star, a complete graph, or an edgeless graph. Because G is connected, G is thus a star or a complete graph. Proposition 4.1 then completes the proof. Proposition 4.2 provides a lower bound on Eρ(G) − Eρ(H), especially when |G| = |H| + 1. The next proposition gives an upper bound on this difference. For two sets A and B, let A△B := (A\B)∪(B \A). For two graphs G 1 and G 2 , let G 1 △G 2 be the graph with vertex set V (G 1 )∪V (G 2 ) and edge set E(G 1 )△E(G 2 ). When E 1 ∩ E 2 = ∅ and G = G 1 △G 2 we say G admits a decomposition into G 1 and G 2 .
Proposition 4.4. Let G 1 and G 2 be graphs and G = G 1 △G 2 . Then Proof. For i = 1, 2, let H i be the graph with vertex set V (G) and edge set E(G i ). Then G = H 1 △H 2 and by Proposition 4.2, Eρ(H i ) = Eρ(G i ) for i = 1, 2. Choose a subset S of V (G) uniformly at random and set T :

This implies immediately that
As a result, we deduce The following corollary asserts that the difference between the average cut-ranks of a graph and its complements is bounded by a fixed constant.
Proof. Let K be the complete graph on the vertex set V (G), then G = K△G as well as G = K△G. By Propositions 4.4 and 4.1 we deduce that which together imply the desired inequality.
It can be seen that the lower bound from Proposition 4.2 and the upper bound from Proposition 4.4 are pretty far apart, because they apply for all graphs in general. In many cases, we need the upper and lower bounds on Eρ(G) − Eρ(H) to be close enough. The next propositions provide a tighter upper bound compared to Proposition 4.4 and a tighter lower bound compared to Proposition 4.2, when we have distinctive structures involving false twins, in particular the attached stars.
Proposition 4.6. Let G be a graph in which u 1 , . . . , u k are pairwise false twins, where k ≥ 1.
In particular if d G (u 1 ) = 1 then Proof. Let n := |G|, H := G − u 1 , V := V (G), and . . , k} such that u j ∈ S and so the row vectors corresponding to u 1 and u j in which completes the proof of the proposition.
where F is the connected subgraph of G consisting of all edges incident with v.
We move on to the left hand side inequality. Observe that for every S ⊆ V , we have ρ G (S) ≥ ρ H (S \ T ). If furthermore v ∈ S and T ⊆ S, then in A G [S, V \ S], the row vectors corresponding to T ∩ S \ {v} are all zero vectors, and for every u j ∈ T \ S, the column vector corresponding to u j has only one 1 as its common entry with the row vector corresponding to v. It follows that in A G [S, V \S], this row vector is linearly independent to the other row vectors, and in A G [S\{v}, V \S], the column vectors corresponding to T \ S are all zero vectors. Hence ρ G (S) = ρ H (S \ T ) + 1.
Therefore, due to the symmetry of ρ G , This completes the proof of the proposition.
We now present an inequality between average cut-rank and maximum cut-rank, which is crucial for our explanations in Section 5. The authors would like to thank Alex Scott (personal communication) for suggesting its proof.
Proposition 4.8. The maximum cut-rank of a graph G is at most four times its average cut-rank.
Proof. Let k be the maximum cut-rank of G. Then there are two disjoint subsets Thus Eρ(H) ≥ k/4 and the conclusion follows.

Well-quasi-ordering and forbidden induced subgraphs
In this section, we prove that graphs of bounded average cut-rank are well-quasi-ordered by induced subgraph relation (Theorem 1.1) and discuss some corollaries.
We start with some definitions solely used in this section. In a graph G, two vertices x, y are called twin-equivalent if either x = y or they are twins. It is easy to check the following.
Proposition 5.1. Let G be a graph.
• The relation "twin-equivalent" is an equivalence relation on V (G).
• If x and y are twin-equivalent, then Here is a fundamental result on the rank and the number of pairwise distinct rows of a 0-1 matrix.
Proof. Let r = rank(M ). Then M has a non-singular r × r submatrix, whose columns are indexed by I. Note that |I| = r and each row vector is completely determined by the 0-1 values on the entries in I and therefore M has at most 2 |I| distinct rows.
We present a structural lemma for graphs of maximum cut-rank at most n and of average cutrank less than n + 1. This result is also essential to the proof of Theorem 1.2. For convenience, let be the maximum cut-rank of G.
Thus, there is a subset S of A such that |S| − X > √ m/2; that is, there are fewer than |S| − √ m/2 edges in H having ends and labels in S. Then, by deleting one end for each such edge, we get a subset T of S satisfying |T | > √ m/2 and for every distinct u, v ∈ T , in H the label of uv does not belong to T . This means that for every distinct u, v ∈ T , in G there is a vertex w outside T which is adjacent to only one of u and v, which implies that A G [T, V (G) \ T ] has more than √ m/2 distinct rows. Hence, by Proposition 5.2, can be partitioned into fewer than 2 2n+2 twin-equivalence classes. In particular, if Eρ(G) < n + 1 then by Proposition 4.8 we have max ρ(G) ≤ 4n + 3.
Now we are ready to present the proof of Theorem 1.1. In fact we shall prove that for every real α ≥ 0, the class of graphs of average cut-rank at most α is well-quasi-ordered by the labeled induced subgraph relation.
Proof of Theorem 1.1. Set n := ⌊α⌋. Lemma 5.3 implies that for every graph G of average cut-rank less than n + 1, V (G) can be partitioned into fewer than 2 8n+8 twin-equivalence classes. If X, Y are distinct twin-equivalence classes, then X is either complete or anticomplete to Y . Let π 0 (G) be the quotient graph of G induced by the twin-equivalence relation. We assign to each vertex C of π 0 (G) one of two colors 1, 2 as follows: • color 1 if C is a clique in G, • color 2 if C is an independent set in G. (If C has size 1 then we color arbitrary.) Then there are at most outcomes of π 0 (G) with colors for any G of average cut-rank less than n + 1. Let (Q, ≤) be a well-quasi-order. For any infinite sequence G 1 , G 2 , . . . of Q-labeled graphs of average cut-rank at most α, there is a strictly increasing subsequence G s 1 , G s 2 , . . . such that G s 1 , G s 2 , G s 3 , . . . have the same π (with colors) with n ′ vertices, for some n ′ ≤ 2 8n+8 . We will assign each vertex C of π(G s i ) a weight that is an element of N n ′ × Q <∞ to represent |C| and all labels of vertices in C. Since N n ′ is well-quasi-ordered by Corollary 2.4 and Q <∞ is well-quasi-ordered by ≤ ∞ by Proposition 2.2, there exist i < j such that G s i is a labeled induced subgraph of G s j by Proposition 2.3.
Together with Proposition 2.2, Theorem 1.1 gives the following result which can be viewed as its generalization.
Corollary 5.4. The class of graphs whose components are of bounded average cut-rank is wellquasi-ordered by the induced subgraph relation.
This corollary yields an important implication of Theorem 1.1, which is the following. This corollary implies that various hereditary classes of graphs with bounded parameters are characterized by finitely many forbidden induced subgraphs. We present an application restricting the number of maximal cliques or the number of maximal independent sets. Proposition 5.6. Let n be a positive integer. Then • every graph with at most n maximal cliques has average cut-rank smaller than 2 n − 1; • every graph with at most n maximal independent sets has average cut-rank smaller than 2 n .
Proof. First, let G be a graph with k ≤ n maximal cliques, say V 1 , . . . , V k . For every nonempty subset I of {1, . . . , k} let W I := j∈I V j and observe that W I is a clique in G. Moreover, observe that Thus, every graph with at most n maximal cliques has average cut-rank smaller than 2 n − 1.
Second, let H be a graph with k ≤ n maximal independent sets, say V 1 , . . . , V k . Then H is a graph with k ≤ n maximal cliques V 1 , . . . , V k . By the preceding argument we have Eρ(H) < 2 n − 1, so by Corollary 4.5 we deduce that Eρ(H) < Eρ(H) + 1 < 2 n . Proposition 5.6 and Corollary 5.5 together immediately imply the following.
Corollary 5.7. Let n be a positive integer and C be a hereditary class of graphs such that every component of G ∈ C has at most n maximal cliques or n maximal independent sets. Then C can be characterized by finitely many forbidden induced subgraphs.

Upper bound on the size of induced subgraph obstructions
In this section we prove Theorem 1.2. We start with a direct corollary of Lemma 5.3 and Proposition 5.1.
Corollary 6.1. For every d ≥ 1, if Eρ(G) < n + 1 and G is a graph on at least 2 8n+8 d vertices, then there is an equivalence class of (V (G), ≡ G ) of size larger than d.
In light of Corollaries 3.2 and 6.1, we come to the proof of Theorem 1.2. For convenience, we recall the sequence {x n (ε)} n≥0 defined in Section 1, as follows.
For n = 0, if the deletion of every vertex of G yields a graph with some isolated vertex then E(G) must be a perfect matching, as G has no isolated vertices. Because |G| ≥ x 0 ≥ 5, for v ∈ V (G) we have G − v contains an induced subgraph isomorphic to 2K 2 . It follows by Proposition 4.2 that Eρ(G − v) ≥ Eρ(2K 2 ) = 1 > ε as desired. Thus, we may assume that there is some v ∈ V (G) such that G − v has no isolated vertices. Then by Proposition 4.3, The base case is completed.
For n > 0, assume that the theorem is true for n − 1, we shall show that it is also true for n. Suppose for the sake of contradiction that G has no proper induced subgraph of average cut-rank larger than α = ε + n, then by Proposition 4.4, for v ∈ V (G) we have Eρ(G) < Eρ(G − v) + 1 ≤ ε + n + 1 < n + 2. Because there is an equivalence class C of (V (G), ≡ G ) of size larger than x n−1 − ⌊log(1 − {2 x n−1 ε})⌋ + 2 in G, by Corollary 6.1. Let z ∈ C such that d G (z) = min v∈C d G (v) and let S := C \ {z} as well as H := G − z. Then so S is nonempty. There are three cases to consider, according to Proposition 3.1. Therefore, G is locally equivalent to a graph G ′ satisfying G ′ [C] is an attached star in G ′ , H is locally equivalent to a graph H ′ satisfying H ′ [S] is an attached star in H ′ , and G ′ − C = H ′ − S.

By Proposition 4.7 and Corollary 4.2 we have
thus H ′ −S contains some induced-subgraph-minimal graph of average cut-rank larger than ε+n−1, say F , as an induced subgraph. By the induction hypothesis, F has less than x n−1 vertices, none of which is isolated in F . Then, Eρ(F ) is a rational number larger than ε + n − 1 whose denominator divides 2 x n−1 −1 , so by Corollary 4.2 By Corollary 4.2, Proposition 4.7, and (1) we thus obtain Therefore, since H = G − z, G has a proper induced subgraph of average cut-rank larger than ε + n, a contradiction. This completes the induction step, hence the proof of Theorem 1.2.

Average cut-rank and vertex-minors
7.1. Forbidden vertex-minors. The following corollary is a direct consequence of Theorem 1.1.
Corollary 7.1. Let C be a class of graphs (closed under taking vertex-minors) of bounded average cut-rank. Then this class admits a finite list of graphs G 1 , G 2 , . . ., G m such that a graph G is in C if and only if G has no vertex-minor isomorphic to G j for every j = 1, . . . , m.
Note that because the average cut-rank can be nonintegral, for each α ≥ 0 it is necessary to distinguish the class of graphs of average cut-rank at most α and the class of graphs of average cut-rank smaller than α. Let L ≤α be the class of all graphs H satisfying Eρ(H) > α and any proper vertex-minor of H has average cut-rank at most α, and let L <α be the set of all graphs H satisfying Eρ(H) ≥ α and any proper vertex-minor of H has average cut-rank smaller than α. Then it is easy to check, by Proposition 4.4, that every graph in L ≤α or L <α has average cut-rank smaller than α + 1. We can also easily deduce the following. Proposition 7.2. A graph has average cut-rank larger than (or at least) α if and only if it contains a vertex-minor in L ≤α (or L <α , respectively).
For two graphs G 1 and G 2 , we write G 1 ∼ G 2 if G 1 is isomorphic to a graph locally equivalent to G 2 . It is easy to check that ∼ is an equivalence relation on graphs. For two classes of graphs S 1 and S 2 , we write S 1 ≃ S 2 if for every G ∈ S 1 there is some H ∈ S 2 satisfying G ∼ H and for every H ∈ S 2 there is some G ∈ S 1 satisfying H ∼ G. Then we can easily verify the following.

Proposition 7.3. The relation ≃ is an equivalence relation on all classes of graphs.
By Proposition 7.2, for every α ≥ 0, L ≤α is ≃ to each list of forbidden vertex-minors for average cut-rank at most α, which is an antichain hence finite by Theorem 1.1. Therefore, by Proposition 7.3, every two finite lists of forbidden vertex-minors for average cut-rank at most α are ≃ to each other. Similarly, L <α and the finite lists of forbidden vertex-minors for average cut-rank smaller than α are ≃ to each other.
The next proposition describes interesting properties of L ≤α and L <α .
• L <α \ L ≤α is the class of all graphs without isolated vertices of average cut-rank exactly α.
• If G ∈ L ≤α \ L <α , then G has a proper vertex-minor of average cut-rank exactly α, say H, which satisfies H ∈ L <α and |G| − |H| ≤ 2. If the equality holds then H can be chosen so that H is an induced subgraph of G and G ∼ = H + K 2 .
Proof. Let G ∈ L <α \ L ≤α . Then Eρ(G) ≥ α, so if Eρ(G) > α, G must have a proper vertex-minor, say H, in L ≤α , but then Eρ(H) > α so G ∈ L <α by definition, a contradiction. On the other hand, by Corollary 4.2, if a graph G has average cut-rank α then G ∈ L <α \ L ≤α . Now let G ∈ L ≤α \ L <α . Then G has a proper vertex-minor of average cut-rank at least α, say H, which also must have average cut-rank at most α. Thus Eρ(H) = α, and we may assume that H ∈ L <α by deleting isolated vertices. Since H is a proper vertex-minor of G, there is some G ′ ∈ L ≤α locally equivalent to G so that H is a proper induced subgraph of G ′ . Let It should be noted that when α is a positive integer, L ≤α \ L <α and L <α \ L ≤α are nonempty: By Propositions 4.1 and 4.4, the graph (2α − 1)K 2 + K 1,2 belongs to L ≤α because it has average cut-rank α + 1/4 and every proper vertex-minor of it has average cut-rank at most α; the graph 2αK 2 belongs to L <α and has average cut-rank exactly α, and it is a proper vertex-minor of (2α − 1) Here is a natural question. For which values of α ≥ 0 we have L ≤α ≃ L <α ? In fact, this holds if and only if there is no graph having average cut-rank exactly α, by Proposition 7.4. So the set of all non-satisfied values of α is actually S Eρ , the set of average cut-rank of all graphs. By the definition of average cut-rank, this set is trivially a subset of {p/2 q : p ∈ N 0 , q ∈ N}. So in particular, L ≤α ≃ L <α holds for every irrational α. However, it seems difficult to explicitly describe S Eρ as a set of rational numbers (in particular a subset of {p/2 q : p ∈ N 0 , q ∈ N}). By the well-quasi-ordering by induced subgraph (Theorem 1.1), we deduce the following topological property of S Eρ . Proposition 7.5. For any α ≥ 0 there is some δ α > 0 such that every graph has average cut-rank outside (α, α + δ α ). This implies that S Eρ is not dense in any interval, hence is nowhere dense in [0, ∞).

7.2.
Lower bound on the number of vertex-minor obstructions. In the previous subsection, we proved that for every α ≥ 0, the lists of forbidden vertex-minors for average cut-rank at most α are finite and have the same size; the same happens for the lists of forbidden vertex-minors for average cut-rank smaller than α. We shall show that, there is some universal constant c > 0 such that for any ε ∈ [0, 1) and nonnegative integer n, a list of forbidden vertex-minors for average cut-rank at most (or smaller than) ε + n must contain at least 2 cn log(n+1) graphs. To do so, we construct a set of at least 2 cn log(n+1) vertex-minor-minimal graphs of average cut-rank larger than ε + n, then obtain from this set another set of at least 2 cn log(n+1) vertex-minor-minimal graphs of average cut-rank at least ε + n. Let us start with several notions to make our arguments clearer.
For a graph G, π(G) denotes the quotient graph of G induced by ≡ G . It is not difficult to see that a graph F without isolated vertices is a forest if and only if π(F ) is a forest and every equivalence class of (V (F ), ≡ F ) induces an attached star in F . In this case, let R(F ) be the set of central vertices in the equivalence classes of (V (F ), ≡ F ). Then it is not difficult to check that F [R(F )] is isomorphic to π(F ). We regard π(F ) as a weighted graph by assigning each vertex C of π(F ) the weight |C|.
For two forests F 1 and F 2 , we shall denote π(F 1 ) ∼ = π(F 2 ) if there is an isomorphism keeping weights from π(F 1 ) to π(F 2 ). From the definitions we can deduce the following easily.
The following is another useful characterization of isomorphic forests.   For a graph G, a vertex v ∈ V (G), and an integer k ≥ 0, we denote by G + v K 1,k the graph obtained by making the disjoint union of G and K 1,k and then adding an edge between v and the central vertex of K 1,k . The following lemma is crucial for our construction. Lemma 7.9. Let G ∈ L ≤ε+n and d ≥ 1 be the size of the largest attached star in G. Then there exists a unique positive integer q 1 = q 1 (G) satisfying G + K 1,q 1 ∈ L ≤ε+n+1 , and q 1 ≥ d. Furthermore, for each v ∈ V (G), there exists a unique positive integer Proof. First, we prove that (2) Eρ(G) ≤ ε + n + 2 1−d ≤ ε + n + 1.
Indeed, if d = 1 then for any u ∈ V (G) we have, by Proposition 4.7, If d > 1, then let u be a leaf in an attached star of size d in G. By Proposition 4.6 and the fact that G ∈ L ≤ε+n , we have and (2) is proved. Hence, because Eρ(G) > ε + n, by Proposition 4.1 there is some q 1 ≥ 1 such that for all k ≥ q 1 and for all 1 ≤ k < q 1 Thus, since (by (2) and (4)) we obtain q 1 ≥ d. We show that G + K 1,q 1 ∈ L ≤ε+n+1 . Indeed, let H be a proper vertex-minor of G + K 1,q 1 , then H is the disjoint union of H 1 and H 2 where H 1 is a vertex-minor of G and H 2 is a vertex-minor of K 1,q 1 such that at least one of these two containments is proper. If H 1 is a proper vertex-minor of G, then since G ∈ L ≤ε+n , and if H 2 is a proper vertex-minor of K 1,q 1 , then Thus G + K 1,q 1 ∈ L ≤ε+n+1 . This proves the first claim. Now let v be a vertex of G. By Proposition 4.7, for all k ≥ q 1 , by (3), and for all 1 ≤ k < q 1 − 1, by (4), by Proposition 4.4, Because G + v K 1,q 1 −1 is a proper induced subgraph of G + v K 1,q 1 and the average cut-rank is strictly monotone with respect to the induced subgraph relation by Proposition 4.2, there is a unique In the formation of G ′ := G + v K 1,q 2 , let x be the central vertex of K 1,q 2 that is adjacent to v and S := V (K 1,q 2 ). We show that G ′ ∈ L ≤ε+n+1 . Indeed, suppose for the contrary that H is an elementary vertex-minor of G ′ with V (G ′ ) = V (H) ∪ {u} such that Eρ(H) > ε + n + 1. By Proposition 7.8, H is locally equivalent to one of G ′ − u, (G ′ * u) − u, and (G ′ ∧ uw) − u for any w adjacent to u in G ′ . We may assume without loss of generality that H is one of these graphs. There are three cases to consider.
Similarly we obtain a contradiction. (c) If u = x then H is the disjoint union of G with q 2 isolated vertices, so H and G have the same average cut-rank which is smaller than ε + n + 1, a contradiction. (d) If u ∈ S \ {x} then H = G + v K 1,q 2 −1 which has average cut-rank smaller than ε + n + 1 by the definition of q 2 , a contradiction. ε + n + 1 < Eρ(H) ≤ Eρ(H − S) + Eρ(K 1,q 2 +d G (v) ) < ε + n + 1, a contradiction. (c) If u = x then H * z is isomorphic to G + v K 1,q 2 −1 , thus has average cut-rank smaller than ε + n + 1, a contradiction.
(3) If H = (G ′ ∧ uw) − u, then we may assume that u is not a leaf and not adjacent to a leaf in G ′ , for otherwise w either is the unique neighbor of u in G ′ or can be chosen to be a leaf adjacent to u, and so H = (G ′ ∧ uw) − u ∼ = G ′ − w, returning to the first case. By the same reasoning we may also assume that u is not a leaf in G ′ . There are two subcases to consider.
Here is another consequence of Lemma 7.9.
Proof. Let H := F + K 1,q 1 (F ) . By Lemma 7.9, q 1 (F ) is at least the maximum label in π(F ), so q 1 (F ) + 1 is the largest label in π(H), which implies that q 1 (H) ≥ q 1 (F ) + 1. Also by Lemma 7.9, q 2 (H, v) ≥ q 1 (H) − 1, and thus q 2 (H, v) is at least q 1 (F ), hence at least the maximum label in π(F ). Now we account for the restriction v ∈ R(F ) in the definition of F ε+2n+2 : Because q 2 (H, v) can possibly be equal to q 1 (F ), to deduce Lemmas 7.12 and 7.13 we require the copy of K 1,q 2 (H,v) attached to v lies in a component different from a copy of K 1,q 1 (F ) . Lemma 7.12. For every F ∈ F ε+n , π(F ) has exactly n + 1 vertices, and in π(F ), no positive integer appears more than twice as a label; if some appears twice then they are in different tree components and one of them is the smallest label in its component.
Proof. We proceed by induction on n. When n = 0 the lemma is trivial. Assuming that the lemma is true for n = 2k, we shall show that it is also true for n = 2k + 1 and 2k + 2. Let F ∈ F ε+2k and consider H := F + K 1,q 1 (F ) ∈ F ε+2k+1 . Set S := V (K 1,q 1 (F ) ). By Lemma 7.9, q 1 (F ) is at least the maximum label in π(F ), so the conclusion holds for π(H) because it also holds for π(F ), which is done by the induction hypothesis. Now consider G := H + v K 1,q 2 (H,v) ∈ F ε+2k+2 for v ∈ R(F ). By Corollary 7.11, q 2 (H, v) is at least q 1 (F ) as well as the maximum label in π(F ). So, since v ∈ R(F ), the labels in π(F ) are preserved in π(G), hence by the induction hypothesis the conclusion for π(G − S) indeed holds. Thus, to verify the conclusion for π(G), it is enough to check two (unique) copies of K 1,q 2 (H,v) and K 1,q 1 (F ) in G. But this is easy, since if q 2 (H, v) > q 1 (F ) then we are done, and if q 2 (H, v) = q 1 (F ) then those two copies must be in different components because v ∈ R(F ). Lemma 7.13. For every ε ∈ [0, 1) and n ≥ 0, no two distinct forests in F ε+n are isomorphic.
Corollary 7.14. For every ε ∈ [0, 1) and k ≥ 0, the number of pairwise nonisomorphic graphs in F ε+2k and F ε+2k+1 is To finish the proof of Theorem 1.3 we need one more lemma.
Lemma 7.15. If ε > 0 or n ≥ 1, then every forest F in F ε+n \ L <ε+n has a leaf, say v, whose deletion yields a forest, say H, in L <ε+n of average cut-rank exactly ε + n. Moreover, if v belongs to an equivalence class of size two of (V (F ), ≡ F ) and its unique neighbor has degree two in F then |π(H)| = n; otherwise |π(H)| = n + 1.
Proof. Let F ∈ F ε+n \ L <ε+n . By Corollary 7.10 and Proposition 7.4, F has a proper vertexminor, say H ′ , of average cut-rank ε + n such that H ′ ∈ L <ε+n and |F | − |H ′ | ≤ 2. Moreover, if |F | − |H ′ | = 2 then H ′ can be chosen so that F ∼ = H ′ + K 2 , so F has a component of size two. In this case, by the construction of F ε+n , Lemma 7.9, and Corollary 7.11 we deduce that n ≤ 1. If n = 0 then F ∼ = K 2 , so H ′ is empty, but this is absurd since ε > 0 by hypothesis; if n = 1 then , then by Proposition 7.8 we may assume without loss of generality that H ′ is one of F − x, (F * x) − x, and (F ∧ xy) − x for any y adjacent to x in F .
(1) If H ′ = F − x then since every equivalence class of (V (F ), ≡ F ) has at least two vertices (the construction of F ε , Lemma 7.9, and Corollary 7.11) and H ′ has no isolated vertices, x is necessarily a leaf in F , so we let v = x. (2) If H ′ = (F * x) − x then we may assume that d F (x) ≥ 2, so if y is a leaf adjacent to x then F − y ∼ = H ′ * x ∈ L <ε+n , and we let v = y. (3) If H ′ = (F ∧ xy) − x then if furthermore x is a leaf in F then y is the unique neighbor of x in F , hence isolated in H ′ , a contradiction. So, d F (x) ≥ 2, and since y can be chosen to be any neighbor of x in F , we may assume that y is a leaf adjacent to x. Then F − y ∼ = H ′ ∈ L <ε+n and we let v = y.
So, we have chosen v. Let H := F − v and u be the unique neighbor of v in F . The first part of the lemma is proved.
We come to the second part of the lemma. If v belongs to an equivalence class of size two of (V (F ), ≡ F ) and d F (u) = 2 then the neighbor of u other than v in F , say w, has degree at least two in F . Let C be the equivalence class of (V (F ), ≡ F ) containing w, then C ∪ {u} is an equivalence class of (V (H), ≡ H ). It follows that |π(H)| = |π(F )| − 1 = n by Lemma 7.12.
In the other cases, it is easy to check that |π(H)| = |π(F )| = n + 1. This completes the proof of the lemma.
We are now ready to prove Theorem 1.3.
First consider when S ≃ L ≤ε+n is a finite list of forbidden vertex-minors for average cut-rank at most ε + n. By Corollary 7.10, F ε+n ⊆ L ≤ε+n , and by Lemmas 7.7 and 7.13, no two distinct forests F 1 and F 2 in F ε+n satisfy F 1 ∼ F 2 . Therefore, for every forest F in F ε+n there is some member in S which is ∼ to F and these members are pairwise not ∼ to each other. By Corollary 7.14, Now consider when S ≃ L <ε+n is a finite list of forbidden vertex-minors for average cut-rank smaller than ε + n. We may assume that ε + n > 0. Let {F 1 , . . . , F m } = F ε+n \ L <ε+n .
For every j = 1, . . . , m, by Lemma 7.15, F j has a leaf, whose deletion yields a forest in L <ε+n , say H j , of average cut-rank exactly ε + n. Moreover, |π(H j )| is either n or n + 1 depending on the condition written in the statement of Lemma 7.15.
Claim. For j ∈ {1, . . . , m}, there are, up to isomorphism, at most n + 1 forests F such that there is some leaf in F whose deletion yields H j .
Proof. There are two cases to consider.
(1) |π(H j )| = n. The only way to obtain F from H j is to add a new vertex to H j and join it to some leaf in H j (to create a new equivalence class of size two). Because (V (H j ), ≡ H j ) has n equivalence classes, each of which induces an attached star in H j , there are at most n forests F satisfying the claim. (2) |π(H j )| = n + 1. The only way to obtain F from H j is to add a new vertex to H j and join it to the central vertex of some equivalence class of (V (H j ), ≡ H j ). Because there are n + 1 such equivalence classes, there are thus at most n + 1 forests F satisfying the claim. Hence there are at most n + 1 desired forests F , completing the proof of the claim.
Let G be a graph on the vertex set {1, . . . , m} such that for distinct j, k ∈ {1, . . . , m}, jk ∈ E(G) if H j ∼ H k . For j ∈ {1, . . . , m}, by Lemma 7.7, k ∈ N G (j) if and only if H j ∼ = H k , implying that there is some forest F ′ k isomorphic to F k such that H j can be obtained by deleting some leaf of F ′ k . Because the set {F 1 , . . . , F m } consists of pairwise nonisomorphic forests, by Lemma 7.13, so does the set {F ′ k : k ∈ N G (j)} ∪ {F j }. It follows by the claim that d G (j) ≤ n for all j = 1, . . . , m. Let S be a maximal independent set in G, then every vertex outside of S is adjacent in G to some vertex in S whose degree is at most n. Hence m = |G| ≤ |S| + n|S|, or equivalently |S| ≥ m n+1 . Let T be the disjoint union of F ε+n ∩ L <ε+n and {H j : j ∈ S}. Since S is an independent set in G, for every distinct j, k ∈ S we have H j ∼ H k . This implies, from our construction, that T ⊆ L <ε+n and no two distinct graphs in T are ∼ to each other. Furthermore, no two distinct forests in F ε+n ∩ L <ε+n are ∼ to each other. Therefore, by (5), and the theorem is completely proved.
8. Characterizations of graphs of average cut-rank at most 3/2 We now aim to prove Theorem 1.4. Our plan is to bound the number of connected components and investigating the maximum induced path of every graph locally equivalent to a fixed graph. This approach not only characterizes graphs of average cut-rank at most 3/2 but also reveals L <1 , L ≤1 , L <3/2 , L ≤3/2 (up to ≃). For every graph G, we denote by p(G) the maximum length of a path graph which is a vertex-minor of G.
Recall that for every k ≥ 0, E k is the graph K 1,k+1 with one edge subdivided. Let us compute Eρ(E k ) for every k ≥ 0 in the following proposition. It explains why {3/2− 3/2 k+2 : k ≥ 0} appears in Theorem 1.4. The technique used is similar to that in the proof of Proposition 4.1.
Proof. Let G be a copy of E k . In the construction of G let x be the central vertex of K 1,k+1 , u be the new vertex, and v be the neighbor of u other than x. We have For every S ⊆ V (G) with u, v ∈ S or u, v / ∈ S, we see that ρ G (S) = 0 if S = ∅ or S = V (G) and 1 otherwise, since the column (or row) corresponding to v in A G [S, V (G) \ S] is zero, u is nonadjacent to every vertex except v and x, and x is adjacent to everybody except v. Note that there are 2 k+2 − 2 nonempty proper subsets S of V (G) of this type.
For every S ⊆ V (G) such that exactly one of u, v lies in S, if S or V (G) \ S contains all the remaining vertices then ρ G (S) = 1 (there are four subsets S of V (G) satisfying this condition). Otherwise, there is some vertex w other than x, u, v such that x and w lie in different sets among S and V (G) \ S, and it is easy to see that ρ G (S) = 2 in this case (there are 2 k+2 − 4 subsets S of V (G) of this type). Therefore Graphs of average cut-rank at most 1. Let us prove an essential result. Proposition 8.2. If G is a connected graph having no path of length three as a vertex-minor then G is isomorphic to a star or a complete graph.
Proof. We prove by induction on |G|. If |G| = 1, 2, 3 then this is trivial, because G is connected. For n ≥ 4, assuming that the proposition is true for |G| < n, we shall show that it is also true for |G| = n. Because G is connected, there is some vertex v ∈ V (G) which is a leaf of some spanning tree of G, so G − v is connected. Because G − v, being an induced subgraph of G, has no vertexminor isomorphic to P 4 , by the induction hypothesis we see that G − v is isomorphic to a star or a complete graph.
First, consider the case that G − v is isomorphic to a complete graph. Suppose that there is some u ∈ V (G) \ {v} which is not adjacent to v in G. Since G is connected, there is some w ∈ N G (v) ∩ V (G), and since n ≥ 4, there is some t ∈ V (G) \ {u, v, w}. Let H := G[{u, v, w, t}].
If t ∈ N G (v), then H * v ∧ vw ∼ = P 4 ; if t ∈ N G (v), then H * w ∼ = P 4 . In both cases, G contains a vertex-minor isomorphic to P 4 , which is a contradiction. Thus every vertex in V (G) \ {v} is adjacent to v, hence G is complete.
If G−v is isomorphic to a star, then let u be the central vertex of G−v. Now G * u−v = (G−v) * u is complete and obviously G * u is connected, so from the previous paragraph we deduce that G * u is complete. It follows that G = (G * u) * u is isomorphic to a star. This completes the proof. Lemma 8.3. Every graph without isolated vertices of average cut-rank at most 1 is ∼ to one of 2K 2 and K 1,k for k ≥ 0. Moreover Proof. Let G be a graph of average cut-rank at most 1 or in L <1 ∪ L ≤1 .
We consider when G is connected. If p(G) ≤ 2, by Proposition 8.2 G is isomorphic to a star or a complete graph which has average cut-rank smaller than 1. If p(G) ≥ 3, then Eρ(G) ≥ Eρ(P 4 ) = 9/8, so G ∼ P 4 .
Let us start with the case that G has precisely two components, then since G has no isolated vertices it has an induced subgraph isomorphic to 2K 2 which has average cut-rank 1, and if moreover G has at least five vertices then it has a vertex-minor isomorphic to K 2 + P 3 which has average cut-rank 7/4. In this case if Eρ(G) ≤ 1 or G ∈ L <1 then G ∼ = 2K 2 , and if G ∈ L ≤1 then G ∼ K 2 +P 3 .
If G has at least three components then it has an induced subgraph isomorphic to 3K 2 which has average cut-rank 3/2. Hence there is no such G of average cut-rank at most 1 or in L <1 , and if G ∈ L ≤1 then G ∼ 3K 2 .

8.2.
Graphs of average cut-rank at most 3/2. Let us start with several technical results.
Lemma 8.4. Let P be an induced path of length three in a graph G and v be a vertex of G outside P such that v has at least two neighbors in P . Then • If v is adjacent to both ends of P , G contains a cycle of length five as a vertex-minor.
• Otherwise, G contains a path of length four as a vertex-minor.
Proof. Let P = abcd and v be a vertex outside P in G. If v is adjacent to a and d then by taking local complementations at a and d if necessary, we see that G ∼ C 5 . Otherwise, we may assume that v is nonadjacent to d.
If v is adjacent to exactly two of a, b, c, then G * v ∼ = P 5 or G ∧ ab ∼ = P 5 ; if v is adjacent to all of a, b, c, then G * a ∧ ab ∼ = P 5 .
Proposition 8.5. Every graph without isolated vertices on five vertices is ∼ to one of K 2 + P 3 , K 1,4 , P 5 , E 2 , and C 5 .
Proof. Let G be a graph on five vertices. If G is disconnected then since G has no isolated vertices, G ∼ K 2 + P 3 . Consider when G is connected. By Proposition 8.2, if p(G) ≤ 2 then G ∼ K 1,4 . If p(G) ≥ 4 then G ∼ P 5 . If p(G) = 3, then G contains an induced path of length three, say P := abcd. By Lemma 8.4, if v is adjacent to both a and d or v is adjacent to at least two of a, b, c, d then G ∼ C 5 or G ∼ P 5 . It remains to check if v is adjacent to exactly one of a, b, c, d, which leads to G ∼ = P 5 or G ∼ = E 2 , the first case fails since p(G) = 4.
Corollary 8.6. If G is a graph on at most five vertices and Eρ(G) ≥ 3/2, then G ∼ C 5 .
Proof. Because the values of the average cut-rank of P 6 , P 4,1 , P 5,1 , P 5,2 , C 3,1 , and C 4,1 are pairwise distinct, by Corollary 4.2 none of them is ∼ to the other. Moreover, these graphs are distancehereditary, hence each of them has no vertex-minor isomorphic to C 5 . The conclusion follows by Corollary 8.6.
Proof. Let G be a graph such that either Eρ(G) ≤ 3/2 or G ∈ L <3/2 ∪ L ≤3/2 . When G has exactly two components, if every component of G has at least three vertices then G has a vertex-minor isomorphic to 2P 3 whose average cut-rank is 3/2, and if furthermore G has at least seven vertices then G has a vertex-minor isomorphic to P 3 + K 1,3 or P 3 + P 4 whose average cut-rank is 13/8 or 15/8, respectively; if one component of G has only two vertices then G = K 2 + H for some graph H with Eρ(H) = Eρ(G) − 1/2. By applying Lemma 8.3 to H, we see that if Eρ(G) ≤ 3/2 then G is ∼ to K 2 + K 1,k for some k ≥ 1 or 2P 3 , if G ∈ L <3/2 then 2P 3 , and if G ∈ L ≤3/2 then G is ∼ to P 3 + K 1,3 or P 3 + P 4 .
If G has at least four components, then G has an induced subgraph isomorphic to 4K 2 which has average cut-rank 2. Thus there is no such G of average cut-rank at most 3/2 or in L <3/2 , and if G ∈ L ≤3/2 then G ∼ = 4K 2 . Now we consider when G is connected. By Lemma 8.3, we may assume that G has average cut-rank larger than 1, so by Proposition 8.2, p(G) ≥ 3.
If p(G) ≥ 5 then G has a vertex-minor isomorphic to P 6 , so if G ∈ L <3/2 ∪ L ≤3/2 then G ∼ P 6 and there is no such G of average cut-rank at most 3/2.
If p(G) = 4, then assume without loss of generality that G contains an induced path of length four, say P = abcde. If G has five vertices then G is isomorphic to P 5 or C 5 . If G has at least six vertices, then there should be a vertex v outside P satisfying N G (v) ∩ V (P ) = ∅. Note that {a, d}, {b, c} ⊆ N G (v) for otherwise C 5 is isomorphic to a proper vertex-minor of G by Lemma 8.4, contradicting the original assumption on G. If {a, e} ⊆ N G (v) then b, d / ∈ N G (v). Pivoting ab if necessary, we see that G has an induced cycle of length six and then a cycle of length five as a proper vertex-minor, which is absurd. Thus at least one of a, e is not adjacent to v, say e. If a ∈ N G (v) then d, e / ∈ N G (v), then by taking local complementation at a and/or pivoting ab if necessary, we have that G has a path of length five as a vertex-minor, which contradicts p(G) = 4. Therefore a, b / ∈ N G (v). If v is adjacent to exactly one of b, c, d, then necessarily G ∼ P 5,1 or G ∼ P 5,2 ; if v is adjacent to precisely two of them then by taking local complementation at v if necessary, we have G ∼ C 4,1 or G ∼ P 6 , the second case fails as p(G) = 4; if v is adjacent to all b, c, d then C 3,1 is isomorphic to a vertex-minor of G * v so G ∼ C 3,1 . Hence, if p(G) = 4 and Eρ(G) ≤ 3/2 then G ∼ P 5 , and if p(G) = 4 and G ∈ L <3/2 ∪ L ≤3/2 then G is ∼ to one of C 5 , P 5,1 , P 5,2 , C 4,1 , and C 3,1 , by Corollary 8.7.
If p(G) = 3 then G contains an induced path of length three, say P = abcd. Then by Proposition 8.5, S := N G (P ) \ V (P ) = ∅. Pick v ∈ S. If {a, d} ⊆ N G (v) then G ∼ C 5 , impossible since p(G) = 3. Otherwise, suppose that d ∈ N G (v). Then by Lemma 8.4 v must be adjacent to only one of b, c. Hence, each vertex in S should be adjacent to only one of b, c. If all vertices in S are pairwise nonadjacent and adjacent to the same among b, c, then G ∼ = E k for some k ≥ 1 and thus Eρ(G) < 3/2. Otherwise, there are two vertices in S, say u, v, being adjacent to each other or adjacent to different vertices in {b, c}. In the case u, v are adjacent, if they are adjacent to the same among b, c then P 5,1 is isomorphic to an induced subgraph of G * u which contradicts p(G) = 3, otherwise G ∧ uv has an induced path of length five which is impossible; in the case u, v are adjacent to different vertices in {b, c}, we only have to check when uv / ∈ E(G), then G ∼ P 4,1 . Thus, if p(G) = 3 and Eρ(G) ≤ 3/2 then G ∼ E k for some k ≥ 0, and if p(G) = 4 and G ∈ L <3/2 ∪ L ≤3/2 then G ∼ P 4,1 , by Corollary 8.7.

Concluding remarks
The tree-depth of a graph G, denoted by td(G), is defined recursively as follows (see [ where G 1 , . . . , G k are connected components of G and k ≥ 2 in the last case. As mentioned in Section 1, Ding [9] proved that graphs of bounded tree-depth are well-quasi-ordered by the induced subgraph relation. Theorem 1.1 shows that graphs of bounded average cut-rank are well-quasiordered by the induced subgraph relation. It is natural to ask whether classes of bounded tree-depth are contained in or contain classes of bounded average cut-rank. We show that this is not the case in both directions. The complete graph K n has tree-depth n and the average cut-rank at most 1 for all n. This shows that not every class of graphs of bounded average cut-rank has bounded tree-depth. For the converse, nK 2 has tree-depth 2 and yet the average cut-rank is n/2. So this example shows that not every class of graphs of bounded tree-depth has bounded average cut-rank. The following proposition shows that in fact, the average cut-rank of a graph is always bounded from above by an increasing function of its strong matching number and its tree-depth. The strong matching number of a graph G, denoted by ν s (G), is the size of a maximum induced matching of G. Let p s (G) be the length of a longest path in G.