Isotropic matroids II: Circle graphs

We present several characterizations of circle graphs, which follow from Bouchet's circle graph obstructions theorem.


Introduction
Let F be a 4-regular graph and let C be an Euler system of F , i.e., a set that includes precisely one Euler circuit of each connected component of F . Then the interlacement graph I(C) is the simple graph with vertex-set V (F ), in which v i and v j are adjacent if and only if they are interlaced with respect to C, i.e., they appear in the order v i ...v j ...v i ...v j or v j ...v i ...v j ...v i on one of the circuits of C. A simple graph that arises from this construction is called a circle graph.
The idea of interlacement is almost 100 years old, as it was used by Brahana in defining his separation matrix [14]. Interlacement graphs were first discussed by Zelinka [43], who credited the idea to Kotzig. But circle graphs did not become well known until the 1970s, when Cohn and Lempel [20] and Even and Itai [21] used them to analyze permutations, and Bouchet [1] and Read and Rosenstiehl [34] used them to study Gauss' problem of characterizing generic self-intersecting curves in the plane. Circle graphs were studied intensively during the next few decades. Among the notable results of this intensive study are polynomial-time recognition algorithms due to Bouchet [3], Gioan, Paul, Tedder and Corneil [26], Naji [31] and Spinrad [35].
See Figure 1 for an example. On the left is a 4-regular graph, with an Euler circuit indicated by dashes. To trace the Euler circuit just walk along the edges, making sure to preserve the dash style (dashed or plain) when traversing Figure 1: An Euler circuit and its interlacement graph. a vertex; the dash style will sometimes change in the middle of an edge, though. On the right is the resulting interlacement graph.
If v ∈ V (F ) then the κ-transform C * v is the Euler system obtained from C by reversing one of the v-to-v walks within the circuit of C incident at v. As we do not distinguish between circuits that differ only in starting point or orientation, the same Euler system will result no matter which of the two vto-v walks is reversed. The κ-transformations were introduced by Kotzig [29], who proved the fundamental theorem that any two Euler systems of F are connected through some finite sequence of κ-transformations. As noted by Read and Rosenstiehl [34], the interlacement graph I(C * v) is the simple local complement I(C) v s , the simple graph obtained from I(C) by reversing all adjacencies among neighbors of v. (We use the subscript s to distinguish this operation from local complementation on looped simple graphs, which also reverses loops in the open neighborhood N (v).) Simple graphs that can be obtained from each other through local complementations are said to be locally equivalent, and an induced subgraph of a simple graph locally equivalent to G is called a vertexminor of G. Bouchet [7] gave a famous obstruction characterization: a simple graph is a circle graph if and only if none of the three graphs pictured in Figure  2 is a vertex-minor. We refer to this famous result as Bouchet's theorem.
Bouchet's theorem resembles several well-known forbidden minors characterizations of matroid classes: for instance a matroid is binary iff U 2,4 is not a minor, a binary matroid is regular iff neither F 7 nor F * 7 is a minor, and a regular matroid is graphic iff neither M * (K 5 ) nor M * (K 3,3 ) is a minor. But Bouchet's theorem involves induced subgraphs rather than matroid minors, and including local equivalence makes Bouchet's theorem seem more complicated than the classic matroid results. The present paper was initially motivated by a couple of questions suggested by this resemblance: Can Bouchet's theorem be rephrased to characterize circle graphs using matroids? If so, is it possible to do so without mentioning local equivalence? It turns out that both answers are "yes"; we present several such characterizations below. In the process of explaining them we also obtain other circle graph characterizations, some of which involve local equivalence and do not explicitly mention matroids.
To state these characterizations, we introduce some terminology. If G is a simple graph with adjacency matrix A then consider the |V (G)| × 3 |V (G)| matrix IAS(G) = I A I + A , with entries in GF (2). Notation for the columns of IAS(G) follows this scheme: is a subset that includes precisely one element of each vertex triple, and a subset of a transversal is a subtransversal. A transverse matroid of G is a matroid obtained by restricting M [IAS(G)] to a transversal. (We use "transverse matroid" to avoid confusion with transversal matroids.) A transverse circuit of G is a circuit of a transverse matroid of G.
The general theory of isotropic matroids is presented in [19] and [39]. Part of this theory is the following result: Theorem 1 [19] Let G and H be simple graphs. Then any one of the following conditions implies the others: In particular, if G and H are locally equivalent then each sequence of local complementations that may be used to obtain H from G yields an induced isomorphism M [IAS(G)] → M [IAS(H)], which is compatible with the partitions of W (G) and W (H) into vertex triples.
All the interlacement graphs of Euler systems of a particular 4-regular graph F are locally equivalent; it follows that the class of circle graphs is closed under local complementation. Theorem 1 then implies that there must be matroidal characterizations of circle graphs using their isotropic matroids, their transverse circuits and their transverse matroids. Circle graph characterizations involving isotropic matroids are complicated by the fact that the class of isotropic matroids is not closed under matroid minors. (The order of an isotropic matroid is divisible by 3, so deleting or contracting an element of an isotropic matroid cannot yield another isotropic matroid.) In order to derive such characterizations we need a special minor operation that is appropriate for isotropic matroids.
Definition 2 Let G be a looped simple graph, let S be a subtransversal of W (G), and let S ′ contain the other 2 |S| elements of W (G) that correspond to the same vertices of G as elements of S. Then the isotropic minor of G obtained by contracting S is the matroid We use the term isotropic minor because the definition is consistent with Bouchet's definitions of minors of isotropic systems [2] and multimatroids [9].
Theorem 3 [39] The isotropic minors of G are precisely the isotropic matroids of vertex-minors of G.
Bouchet's theorem now leads directly to a characterization of circle graphs by excluded isotropic minors.
Theorem 4 Let G be a simple graph. Then G is a circle graph if, and only if, the isotropic matroids of W 5 , BW 3 and W 7 are not isotropic minors of G.
It follows that we may try to gain insight into the special characteristics of circle graphs by contrasting their isotropic minors of size ≤ 24 with the isotropic matroids of W 5 , BW 3 and W 7 . Formulating and verifying these contrasts is facilitated by the following four theorems, which show that isotropic matroids reflect important structural properties of graphs.
Theorem 5 Let G be an interlacement graph of an Euler system of a 4-regular graph F , and let k be a positive integer. If F has a k-circuit then G has a transverse circuit of size ≤ k.
Theorem 5 is discussed in Section 4. The next three theorems are discussed in [19].
Theorem 6 [19] Let G be a simple graph, and let k be a positive integer. Then G has a transverse k-circuit if and only if some graph locally equivalent to G has a vertex of degree k − 1.
Theorem 7 [19] Let G be a simple graph, and let k 1 , k 2 be positive integers. Then G is locally equivalent to a graph with adjacent vertices of degrees k 1 − 1 and k 2 − 1 if and only if G has transverse circuits γ 1 , γ 2 such that |γ i | = k i , the largest subtransversals contained in γ 1 ∪ γ 2 are of cardinality |γ 1 ∪ γ 2 | − 2, and two of these largest subtransversals are independent sets of M [IAS(G)].
Theorem 8 [19] Let G be a simple graph, and let k 1 , k 2 be positive integers. Then these two conditions are equivalent: • G is locally equivalent to a graph with nonadjacent vertices of degrees k 1 −1 and k 2 − 1, which share no neighbor.
Here is an illustration of the usefulness of these properties. It is easy to see that up to isomorphism, there are only two simple 4-regular graphs with ≤ 6 vertices: one is K 5 and the other is obtained from K 6 by removing the edges of a perfect matching. Each of these graphs contains several 3-circuits. A non-simple 4-regular graph must contain a 1-circuit or a 2-circuit, of course, so Theorem 5 tells us that every circle graph with ≤ 6 vertices has a transverse circuit of size ≤ 3. According to Theorem 6, this is equivalent to saying that every circle graph with ≤ 6 vertices is locally equivalent to a graph with a vertex of degree ≤ 2. Inspecting the matrix IAS(W 5 ), it is not hard to see that the only circuits of size ≤ 3 in M [IAS(W 5 )] are vertex triples; the smallest transverse circuits are of size 4. It follows from Theorem 6 that no simple graph locally equivalent to W 5 has a vertex of degree ≤ 2. This is enough to verify the following.
Corollary 9 Let G be a simple graph with ≤ 6 vertices. Then any one of the following properties implies the others.
1. G is a circle graph.
2. G has a transverse circuit of size ≤ 3.
3. G is locally equivalent to a graph with a vertex of degree ≤ 2.
Further analysis leads to several different characterizations of larger circle graphs. For instance, here is a characterization that involves both isotropic matroids and transverse matroids.  2. Every isotropic minor of G of size < 24 has a loop or a pair of intersecting 3-circuits.
3. Suppose an isotropic minor M of G of size 24 has no loop and no pair of intersecting 3-circuits. Then every transverse matroid of M that contains two disjoint circuits also contains other circuits.
The three conditions of Theorem 10 correspond directly to the three obstructions of Bouchet's theorem: condition 1 excludes BW 3 , condition 2 excludes W 5 , and condition 3 excludes W 7 . By the way, we state condition 2 this way only for variety. As vertex triples are dependent sets, it is not hard to see that an isotropic matroid has a transverse circuit of size ≤ 3 if and only if it has a loop or a pair of intersecting 3-circuits. Details of the argument appear in the proof of Corollary 40.
Condition 1 is of particular interest for several reasons. One reason is simply that the cographic property is more familiar than the small-circuit properties mentioned in conditions 2 and 3. Another reason is that it is possible to explicitly construct the graphs whose cocycle matroids are the transverse matroids of a circle graph; see Section 4 for details. Yet another reason is that as we show in Section 8, condition 1 suffices to characterize a special type of circle graph.
Theorem 11 Let G be a simple graph that is locally equivalent to a bipartite graph. Then any one of the following properties implies the others.
1. G is a circle graph.
2. Every transverse matroid of G is cographic. Figure 3 is a vertex-minor of G.

Neither graph of
Bipartite circle graphs are of particular interest because a simple argument shows that all circle graphs are vertex-minors of bipartite circle graphs. Details are given in Section 9, along with some results about the connection between the crossing number of a 4-regular graph and the matroidal properties of its associated circle graphs. Returning to the general case, note that conditions 2 and 3 of Theorem 10 indicate characteristic properties of the transverse circuits of small isotropic minors of circle graphs. Using Theorem 6, we may reformulate these properties by saying that low-degree vertices are distributed in a characteristic way in small vertex-minors of circle graphs. To be precise: Theorem 12 Let G be a simple graph, and let VM 8 (G) denote the set of graphs with 8 or fewer vertices, which are vertex-minors of G. Then G is a circle graph if and only if every H ∈ VM 8 (G) satisfies at least one of the following conditions.
1. Some graph locally equivalent to H has a vertex of degree 0 or 1.
2. Some graph locally equivalent to H has a pair of adjacent degree-2 vertices.
3. Every graph locally equivalent to H has a vertex of degree 5.
Notice that as every H ∈ VM 8 (G) must satisfy one of the conditions, condition 3 is logically equivalent to the simpler requirement that H itself must have a vertex of degree 5. Also, condition 3 may be replaced by the weaker requirement that there be a vertex of degree ≥ 4, because W 5 and W 7 are both locally equivalent to 3-regular graphs. See Figure 4.
Several other circle graph characterizations are presented in Sections 5 -9. Although the details differ, most are variations on the theme "circle graphs have vertex-minors with distinctive distributions of small transverse circuits." Before deriving these matroidal characterizations of circle graphs, we discuss a different kind of structural characterization of circle graphs, using deltamatroids. It is shown in [24] that a variant of unimodularity called principal unimodularity precisely corresponds to representability of delta-matroids D over every field. This specializes to the usual notion of unimodularity in case D is a matroid. In this way, this result generalizes the well-known result that unimodular representations of matroids correspond precisely to matroids representable over every field. It has been shown by Bouchet (see Geelen's PhD thesis [24]) that circle graphs are precisely the graphs G such that for every graph G ′ locally equivalent to G, G ′ allows for a principal unimodular representation. In Section 2 we reformulate this characterization in terms of binary delta-matroids. Using this characterization, we provide in Section 3 a characterization of isotropic matroids of circle graphs in terms of principal unimodularity, but without mentioning local equivalence. The techniques used in Section 3 are from multimatroid theory [8] and it is essentially shown that the natural 3-matroid generalization of principal unimodularity precisely characterizes isotropic matroids of circle graphs.
More specifically, for an isotropic matroid M [IAS(G)] and transversal T , T has a representation E such that for each transversal T ′ disjoint from T , the determinant of the matrix obtained from E by restricting to the columns of T ′ is equal to 0, 1, or −1.
We show the following (cf. Theorem 25).

Characterizing circle graphs by delta-matroid regularity
In this section we recall a characterization of circle graphs using the notion of regularity for delta-matroids and multimatroids.

Delta-matroids
A set system (over V ) is a tuple D = (V, S) such that S ⊆ 2 V is a set of subsets of a ground set V of D. For notational simplicity we write X ∈ D to denote X ∈ S. We say that D is empty if S = ∅. A delta-matroid D is a nonempty set system (V, S) that satisfies the following property: for all X, Y ∈ S and x ∈ X ∆ Y , there is a y ∈ X ∆ Y (we allow y = x) such that X ∆{x, y} ∈ S [4]. It turns out that if all sets of D have the same cardinality, then D is a matroid represented by its bases [4]. In this way, a delta-matroid can be viewed as a generalization of the notion of matroid. For X ⊆ V , we define the twist of D by X as the set system It turns out that D * X is a delta-matroid if and only if D is a delta-matroid. A delta-matroid D is called even if the cardinalities of the sets of D have equal parity.

Representable and regular delta-matroids
For finite sets X and Y , an X × Y -matrix A is a matrix where the rows and columns of A are indexed by X and Y , respectively. If W ⊆ X and W ⊆ Y , then A[W ] denotes the W × W -matrix obtained from A by removing the entries outside W . We now fix a finite set V . A V ×V -matrix A over some field F is said to be skew-symmetric if −A T = A (note that we allow nonzero diagonal entries in case F is of characteristic 2). If A is skew-symmetric, then is nonsingular} is a delta-matroid [4]. For a skew-symmetric V ×V -matrix A over F, D A is even if and only if all diagonal entries of A are zero.
(In particular, if A is principally unimodular, then each entry of A is equal to 0, 1, or −1.) We say that D is regular if D is representable by a skew-symmetric principally unimodular matrix over R. The following result is shown by Geelen [24]. Theorem 14 ([24]) Let D be an even delta-matroid. Then the following statements are equivalent.
The notion of regularity for delta-matroids generalizes the notion of regularity for matroids. Recall that a matroid M is called regular if M is representable by a totally unimodular matrix over R, where a matrix B over R is said to be totally unimodular if the determinant of every submatrix of B is equal to 0, 1, or −1. We may assume that B is in standard form (I E). Now, it is easy to verify that X Y X I E is totally unimodular if and only if the skew-symmetric matrix V × V -matrix [12]. If A is a skew-symmetric matrix over GF (2) (equivalently, A is symmetric over GF (2)), then D A uniquely determines A [13].
. We say that D is binary if D is representable over GF (2).

Eulerian delta-matroids
A delta-matroid D is said to be Eulerian if D = D A(G) * X where G is a circle graph and X ⊆ V (G) [24]. For notational convenience, this definition is slightly different from [24] as there it is required that X = ∅. The next lemma shows that this difference is not essential. Proof. The only if direction is trivial. To prove the converse, let D = D A(G) be Eulerian. Then D * X = D A(G ′ ) for some circle graph G ′ . Hence, D A(G) = D A(G ′ ) * X. It is shown in [4] that this implies that G is locally equivalent to G ′ . By Theorems 1 and 4, G is a circle graph as well.
It follows from de Fraysseix [23] that Eulerian delta-matroids are a generalization of planar matroids (i.e., cycle matroids of planar graphs); see also [24,Theorem 4.16]. Since D A uniquely determines A, a characterization of Eulerian delta-matroids directly implies a characterization of circle graphs. The following characterization of Eulerian delta-matroids is from [24].

Theorem 17 ([24])
Let D be an even binary delta-matroid, i.e., D = D A(G) * X for some simple graph G and X ⊆ V (G). Then D is Eulerian if and only if, for every graph G ′ locally equivalent to G, D A(G ′ ) is regular.
In particular, every Eulerian delta-matroid is regular. The converse does not hold: any regular matroid M that is not planar is a counterexample by Theorem 16. So, e.g., the cycle matroids of K 3,3 and K 5 are regular, but they are not Eulerian delta-matroids.
Since D A(G) uniquely determines G, it is natural to formulate the notion of local equivalence for binary delta-matroids. For this we require an additional operation on delta-matroids. For a delta-matroid D over V and X ⊆ V , loop complementation of D by X, denoted by D + X, is defined by Y ∈ D + X if and only if there are an odd number of Z ∈ D with Z ⊆ Y . It turns out that the family of binary delta-matroids is closed under loop complementation.
We now use loop complementation to reformulate Theorem 17.
Theorem 18 Let D be an even binary delta-matroid over V . Then the following statements are equivalent.
3. every even delta-matroid obtainable from D by applying a sequence of + and * operations is regular.
Proof. It is shown in [15,Theorem 12] that any delta-matroid D ′ obtainable from D by applying a sequence of + and * is of the form and only if D * X + Y is even (regular, resp.). Hence the last two statements are equivalent. Also, it is shown in [15,Theorem 27] that for simple graphs G and G ′ , G ′ is locally equivalent to G if and only if D A(G ′ ) can be obtained from D A(G) by applying a sequence of + and * operations. By Theorem 17, we obtain that statement (3) implies that D is Eulerian. Conversely, let D be Eulerian. Then D = D A(G) * X for some simple graph G and X ⊆ V (G). Let ϕ be a sequence of + and * operations such that Dϕ is even. Hence G ′ is locally equivalent to G and so D A(G ′ ) is regular (as D is Eulerian). Thus Dϕ = D A(G ′ ) * Y is regular and so statement (3) holds.

Corollary 19
Let M be a binary matroid. Then M is planar if and only if every even delta-matroid obtainable from D by applying a sequence of + and * operations is regular.

Characterizing circle graphs by multimatroid regularity
In this section we reformulate Theorem 17 in terms of isotropic matroids.

Sheltering matroids
First we recall some notions and notation for the theory of multimatroids [8].
Let Ω be a partition of a finite set U . A T ⊆ U is called a transversal (subtransversal, resp.) of Ω if |T ∩ ω| = 1 (|T ∩ ω| ≤ 1, resp.) for all ω ∈ Ω. We denote the set of transversals of Ω by T (Ω) and the set of subtransversals of Ω by S(Ω). A p ⊆ U is called a skew pair of ω ∈ Ω if |p| = 2 and p ⊆ ω. We say that Ω is a q-partition if q = |ω| for all ω ∈ Ω.
Multimatroids form a generalization of matroids. Like matroids, multimatroids can be defined in terms of rank, circuits, independent sets, etc. Here they are defined in terms of independent sets.

Definition 20 ([8])
Let Ω be a partition of a finite set of U . A multimatroid Z over (U, Ω), described by its independent sets, is a triple (U, Ω, I), where I ⊆ S(Ω) is such that: 1. for each T ∈ T (Ω), (T, I ∩ 2 T ) is a matroid (described by its independent sets) and 2. for any I ∈ I and any skew pair p = {x, y} of some ω ∈ Ω with ω ∩ I = ∅, If Ω is a q-partition, then we say that Q is a q-matroid. A basis of Z is a set in I maximal with respect to inclusion. For Note that tight multimatroids are necessarily nondegenerate.
We now consider the related notion of sheltering matroid introduced in [19]. It is a generalization of the matroid M [IAS(G)].

Definition 21
A sheltering matroid is a tuple Q = (M, Ω) where M is a matroid over some ground set U and Ω is a partition of U such that for any independent set I ∈ S(Ω) of M and for any skew pair p = {x, y} of ω ∈ Ω with ω ∩ I = ∅, I ∪ {x} or I ∪ {y} is an independent set of M .
It is shown in [19] that (M [IAS(G)], Ω) is a sheltering matroid with Ω the set of vertex triples of G.
Matroid notions carry over straightforwardly to sheltering matroids. For example, for X ⊆ U , we define the deletion of X from Q by Note that if Q = (M, Ω) is a sheltering matroid, then Z(Q) = (U, Ω, I) with U the ground set of M and I = {I ∈ S(Ω) | I is an independent set of M } is a multimatroid. We say that Z(Q) is the multimatroid corresponding to Q. Also, we say that M shelters the multimatroid Z(Q). Not every multimatroid is sheltered by a matroid [8]. If Z(Q) is a q-matroid, then Q is called a q-sheltering matroid. Moreover, Q is called tight when Z(Q) is tight.
Let Q 1 = (M 1 , Ω 1 ) and Q 2 = (M 2 , Ω 2 ) be sheltering matroids. An isomorphism ϕ from Q 1 to Q 2 is an isomorphism from M 1 to M 2 that respects the skew classes, i.e., if x and y are elements of the ground set of M 1 , then x and y are in a common skew class of Ω 1 if and only if ϕ(x) and ϕ(y) are in a common skew class of Ω 2 .
For a q-partition Ω (of some finite set U ), a transversal q-tuple of Ω is a sequence τ = (T 1 , . . . , T q ) of q mutually disjoint transversals of Q. A projection π of Ω is a function U → V with |V | = |Ω| such that π(x) = π(y) if and only if x, y ∈ ω for some ω ∈ Ω.

Representable and Regular 2-Sheltering Matroids
Note: for notational convenience, from now on we often assume a given fixed 2-partition Ω of some finite set U and a transversal 2-tuple τ = (T 1 , T 2 ) of Ω.
We say that a matrix B over some field F represents the 2-sheltering matroid We now define a notion of regularity for 2-sheltering matroids. We say that a 2-sheltering matroid Q = (M, Ω) is t-regular if Q has a representation B = I A over R such that A is skew-symmetric and for each T ∈ T (Ω), the determinant of the matrix obtained from B by restricting to the columns of T is equal to 0, 1, or −1. Similarly, we say that a 2-matroid Z is t-regular if there is a t-regular 2-sheltering matroid Q such that Z(Q) = Z.
Note that t-regularity for a 2-sheltering matroid (M, Ω) is not the same as regularity of M . On the one hand, it can happen that M is regular but (M, Ω) is not t-regular, if no representation of M has the required skew symmetry with respect to Ω. See [19, Section 2.3] for an example. On the other hand, it can happen that (M, Ω) is t-regular but M is not regular. For example, let M be represented by the matrix Then B satisfies the definition of t-regularity with respect to Ω. The minor M/{t 1,1 , t 1,3 } is represented by the matrix obtained from B by removing the first and third rows and columns: We now make the following observation.
Then A is principally unimodular if and only if, for each transversal T of Ω, the determinant of the matrix obtained from V V I A by restricting to the columns of T is equal to 0, 1, or −1.

Eulerian and t-regular 3-Sheltering Matroids
Note: for notational convenience, from now on we often assume a given fixed 3-partition Ω of some finite set U and a transversal 3 The following lemma is a straightforward reformulation of [39,Theorem 41].
The main result of this section is as follows and is proved in the next subsection.
Theorem 25 Let Q be an isotropic 3-sheltering matroid. Then Q is Eulerian if and only if Q is t-regular.

Proof of Theorem 25
To prove Theorem 25 we translate Theorem 18 from delta-matroids to sheltering matroids.
We recall the following results from [8].
is a one-to-one correspondence from the family of delta-matroids D over V to the family of 2-matroids over (U, Ω).
Moreover, it is shown in [10] that D is even if and only if Z(D, τ, 2) is tight.
By Lemma 22, we have the following.

Lemma 30
For every binary delta-matroid D, the 2-matroid Z(D, τ, 2) is tregular if and only if D is regular.
We say that a 3-matroid Z is t-regular if there is a t-regular 3-sheltering matroid Q such that Z(Q) = Z.

Lemma 33
Let D be a binary delta-matroid. Then D is Eulerian if and only if Z(D, τ, 3) is t-regular (again we assume an arbitrary fixed transversal 3-tuple τ ).
Proof. By Theorem 18, D is Eulerian if and only if every even delta-matroid obtainable from D by applying a sequence of + and * operations is regular.
First assume that Z(D, τ, 3) is t-regular. Let ϕ be a sequence of + and * operations such that Dϕ is even.
The reverse implication is similar. Thereto, assume that D is Eulerian. Let We are now ready to prove Theorem 25.
, τ * X, 3)) for some circle graph G. By Lemma 29, Q = Q(A(G), τ * X, 3) and so Q is Eulerian. Now assume that Q is Eulerian. Then Z(Q) = Z(Q(A(G), τ, 3)) for some circle graph G. Hence D A(G) is Eulerian and by Lemma 33

4-regular graphs
In this section we discuss the relationship between interlacement graphs and circuits of 4-regular graphs.
We begin by establishing some notation and terminology. We think of an edge of a graph as consisting of two distinct half-edges, one incident on each end-vertex of the original edge. A circuit of length ℓ is then a sequence v 1 , h 1 , h ′ 1 , v 2 , h 2 , ..., h ′ ℓ , v ℓ+1 = v 1 of vertices and half-edges, with the property that for each i, h ′ i−1 and h i are half-edges incident on v i and {h i , h ′ i } is an edge incident on v i and v i+1 . Vertices may appear repeatedly on a circuit, but there must be 2ℓ different half-edges. We do not distinguish between circuits that differ only in orientation or starting point. That is, if v 1 , h 1 , h ′ 1 , v 2 , h 2 , ..., h ′ ℓ , v ℓ+1 = v 1 is a circuit then for each index i, the same cir-

The transition matroid
Notice that there are two ways to write a circuit as a sequence of pairs of halfedges: one way is to pair consecutive half-edges of an edge, and the other is to pair consecutive half-edges incident at a vertex. We call each of these latter pairs {h ′ i , h i+1 } a single transition. A transition at a vertex of a 4-regular graph is a pair of disjoint single transitions at the same vertex; the 3 |V (F )|-element set that contains all the transitions of F is denoted T(F ). (Our terminology is slightly nonstandard; for instance Bouchet used "transition" and "bitransition" rather than "single transition" and "transition".) An Euler system C of F gives rise to a notational scheme for T(F ). First, arbitrarily choose an orientation for each circuit of C. Then the three transitions at a vertex v of F may be described as follows: one is used by C, one is not used by C and is consistent with the orientation of the circuit of C incident at v, and the last one is inconsistent with this orientation. (It is easy to see that changing the orientations of some circuits of C will not change the description of any transition.) We label these three transitions φ C (v), χ C (v) and ψ C (v) respectively.
It is important to note that different Euler systems of F give rise to different notational schemes for T(F ). That is, a particular transition τ ∈ T(F ) may be labeled φ for some Euler systems, χ for others, and ψ for the rest. For example, the reader might take a moment to verify that if C and D are the Euler circuits of K 5 indicated on the left and right in Figure 5 (respectively), and v is the vertex at the bottom of the figure, then Given an Euler system C of a 4-regular graph F , we use the notational scheme for T(F ) to associate transitions of F with elements of the isotropic matroid of the interlacement graph I(C). That is, if IAS(I(C)) = I A(I(C)) I + A(I(C)) , where I is an identity matrix and A(I(C)) is the adjacency matrix of I(C), then the v column of I is associated with the transition φ C (v), the v column of A(I(C)) is associated with the transition χ C (v), and the v column of I+A(I(C)) is associated with the transition ψ C (v).
This may to seem to define many different matroids on T(F ), but in fact there is only one: Theorem 34 [40] Let C and D be any two Euler systems of a 4-regular graph F . Then the matrices IAS(I(C)) and IAS(I(D)) represent the same binary matroid on T(F ).
We call the matroid on T(F ) defined by any matrix IAS(I(C)) the transition matroid of F , and denote it M τ (F ). We refer to the transverse circuits, transverse matroids and vertex triples of I(C) as transverse circuits, transverse matroids and vertex triples of M τ (F ).

Circuit partitions and touch-graphs
As mentioned in Theorem 10, all transverse matroids of circle graphs are cographic. The first special case of this property was observed by Jaeger [27], who proved that if G is a circle graph then the binary matroid represented by the adjacency matrix of G -that is, the transverse matroid consisting of all the χ elements of W (G) -is cographic. Jaeger explained this result further in [28], using the core vectors of walks in a graph drawn on a surface. In his papers introducing isotropic systems, Bouchet [2,6] gave a different definition, related to Jaeger's core vectors.

Definition 35
Let F be a 4-regular graph, and P a partition of the edge-set of F into edge-disjoint circuits. The touch-graph T ch(P ) is the graph with a vertex for each circuit of P and an edge for each vertex of F , the edge corresponding to v incident on the vertex or vertices corresponding to circuits of P that are incident at v. Definition 35 is important for us because touch-graphs of circuit partitions of a 4-regular graph F are closely related to transverse matroids of M τ (F ). This connection is given by the following result of [38]. (See also [18,37].) Recall that the vertex cocycle associated with a vertex in a graph is the set of non-loop edges incident on that vertex.
Theorem 36 [38] Let P be a circuit partition of a 4-regular graph F , let C be an Euler system of F , and let M (C, P ) be the submatrix of IAS(I(C)) that includes those columns that correspond to transitions included in P . Then the vertex cocycles of T ch(P ) span the vector space  Figure 6: A circuit partition in K 5 , and its touch-graph.
Equivalently, the binary matroid represented by M (C, P ) is the cocycle matroid of T ch(P ) (the dual of the more familiar circuit matroid). As the matroids represented by the various M (C, P ) matrices are the transverse matroids of I(C), Theorem 36 tells us that condition 1 of Theorem 10 is necessary for a circle graph.
Theorem 36 generalizes and unifies ideas of Bouchet [2,11] and Jaeger [27,28]. Jaeger's core vector theory includes a special case of Theorem 36, which requires (in our notation) that P not follow the φ C (v) transition at any vertex. Bouchet's discussion of graphic isotropic systems includes the cocycle spaces of touch-graphs, but does not include an explicit matrix formulation.
As an example, consider the circuit partition P of K 5 illustrated in Figure 6. The reader may verify that if C and D are the Euler circuits of K 5 indicated in Figure It is a simple matter to verify that the nonzero elements of ker M (C, P ) = ker M (D, P ) are the vertex cocycles of T ch(P ), i.e., the column vectors obtained by transposing (1, 1, 1, 0, 1), (1, 1, 0, 1, 0) and (0, 0, 1, 1, 1).
The following consequence of Theorem 36 allows us to study the transverse circuits of circle graphs using circuits of 4-regular graphs.
Corollary 37 Let γ be a circuit of a 4-regular graph F , which is not an Euler circuit of a connected component. Let τ (γ) denote the set of transitions involved in γ. Then τ (γ) is a dependent set of M τ (F ).   Arbitrarily choose transitions at vertices of V 0 , and let P denote the circuit partition that involves these transitions in addition to the elements of τ (γ). Let C be any Euler system for F . Then Theorem 36 tells us that the vertex cocycle of γ in T ch(P ) is an element of ker M (C, P ). This vertex cocycle is V 1 , considered as a set of edges in T ch(P ).
As V 1 ∈ ker M (C, P ), the columns of M (C, P ) corresponding to elements of V 1 sum to 0. These columns are the columns of IAS(I(C)) corresponding to transitions of P at vertices of V 1 , so they correspond to a subset of τ (γ). As V 1 = ∅, it follows that τ (γ) is dependent in M τ (F ).
Theorem 5 of the introduction follows readily. If γ is not an Euler circuit of a connected component, then Corollary 37 tells us that τ (γ) contains a transverse circuit of M τ (F ). If γ is an Euler circuit of a connected component, then that connected component has a non-Euler circuit γ ′ ; necessarily |τ (γ ′ )| ≤ |τ (γ)|, and Corollary 37 tells us that τ (γ ′ ) contains a transverse circuit of M τ (F ).
By the way, it can happen that a transverse circuit of M τ (F ) is strictly smaller than every circuit in F . See Figure 7 for an example. F is a simple graph, so it has no circuit of size < 3. But the interlacement graph of the indicated Euler circuit C has a transverse circuit of size two: if v is the vertex at the top of the figure and w is the vertex pendant on v in I(C) then the columns of IAS(I(C)) representing φ C (v) and χ C (w) are equal, so {φ C (v), χ C (w)} is a transverse circuit.
Before concluding this section we should explain the connection between isotropic minors of circle graphs (as defined in Definition 2) and detachments of 4-regular graphs. Let F be a 4-regular graph, and suppose τ ∈ T(F ) is a transition at a vertex v. Then the detachment of F along τ is the 4-regular graph F ′ obtained from F by removing v and forming two new edges from the four half-edges incident at v, pairing together the half-edges according to τ . If F has a loop at v, detachment along τ may result in one or two "free edges" that have no incident vertex; any such free edge is simply discarded. If τ is not a loop of M τ (F ), F has an Euler system C with φ C (v) = τ . As illustrated in Figure 8, F ′ then inherits an Euler system C ′ directly from C. Clearly I(C ′ ) is the induced subgraph of I(C) obtained by removing v. Consequently IAS(I(C ′ )) is the matrix obtained from IAS(I(C)) by removing the v row and all three columns corresponding to v. As the only nonzero entry of the φ C (v) column is a 1 in the v row, the effect of these removals is that

Circle graphs with small transverse circuits
Corollary 37 suggests that in order to characterize circle graphs, we should obtain detailed information about small circuits in small 4-regular graphs.
Proposition 38 Let F be a 4-regular graph with < 9 vertices. If F has no circuit of size ≤ 3, then F is isomorphic to K 4,4 .
Proof. Suppose F has no circuit of size < 3; then F is a simple graph. According to data tabulated by Meringer [30], there are only ten simple, 4-regular graphs of order < 9, up to isomorphism. One is K 4,4 . The other nine are pictured in Figure 9. It is evident that all nine have 3-circuits.
Close inspection of Figure 9 yields a more elaborate form of Proposition 38, which will also be useful. In stating this proposition we use a convenient shorthand, specifying a circuit by simply listing the incident vertices in order.

Proposition 39
Let F be a simple 4-regular graph with < 9 vertices, which is not isomorphic to K 4,4 . Then F has two distinct 3-circuits γ 1 = v 1 v 2 w 1 and γ 2 = v 1 v 2 w 2 , and an Euler circuit of the form v 1 w 1 v 2 v 1 w 2 v 2 ... Corollary 40 A simple graph G is a circle graph if and only if G satisfies these two conditions.
1. Every transverse matroid of G is cographic.
2. If an isotropic minor of G of size ≤ 24 does not have a loop or a pair of intersecting 3-circuits, then it is isomorphic to M τ (K 4,4 ).
Proof. Suppose G is an interlacement graph of a 4-regular graph F . Condition 1 follows from Theorem 36. As discussed at the end of Section 4, the isotropic minors of G are isotropic matroids of interlacement graphs of 4-regular graphs obtained from F through detachment. Consequently, if M is an isotropic minor of G of size ≤ 24 then M is the isotropic matroid of a circle graph H with ≤ 8 vertices. Proposition 38 tells us that if H is not an interlacement graph of K 4,4 then it is an interlacement graph of a 4-regular graph with a circuit of size ≤ 3. Theorem 5 tells us that H has a transverse circuit of size ≤ 3.
If Conceptually, condition 1 of Corollary 40 may seem to be more interesting than condition 2, because cographic matroids are well-studied and the fact that transverse matroids of circle graphs are cographic is explained by their connection with touch-graphs. However, we have not been able to formulate a matroidal characterization of circle graphs that references only broad properties like "cographic." Instead, Proposition 39 allows us to replace condition 1 of Corollary 40 with a requirement involving 3-circuits.  Then Proposition 39 tells us that F ′ has two distinct 3-circuits γ 1 = v 1 v 2 w 1 and γ 2 = v 1 v 2 w 2 that share precisely one edge. Corollary 37 tells us that τ (γ 1 ) and τ (γ 2 ) are transverse 3-circuits of M . As γ 1 and γ 2 do not involve the same transition at v 1 or v 2 , no subtransversal of H contains more than four elements of τ (γ 1 ) ∪ τ (γ 2 ). For the converse, suppose G satisfies the statement. Then no isotropic minor of G is isomorphic to M [IAS(W 5 )] or M [IAS(W 7 )], because neither of these isotropic matroids is isomorphic to M τ (K 4,4 ) and neither has any transverse circuit of size ≤ 3. BW 3 does have transverse 3-circuits: the neighborhood circuits of the degree-2 vertices are of size 3, and the χ G elements of the three degree-2 vertices also constitute a transverse circuit. A computer search using the matroid module for Sage [33,36] verifies that these four are the only transverse 3-circuits of BW 3 , though, and clearly their union is a transversal.
Recall the notation used in the introduction: If G is a simple graph, then VM 8 (G) denotes the set of vertex-minors of G with 8 or fewer vertices. Using this notation and the fact that the isotropic minors of G are the isotropic matroids of vertex-minors of G, we may rephrase Corollary 41 as follows. For the converse, suppose G is a circle graph and H ∈ VM 8 (G) is not an interlacement graph of K 4,4 and not locally equivalent to a graph with a vertex of degree 0 or 1. Then H is an interlacement graph of a 4-regular graph F pictured in Figure 9. As noted in Proposition 39, F has an Euler circuit C of the form v 1 w 1 v 2 v 1 w 2 v 2 . . .. Then v 1 and v 2 are adjacent degree-2 vertices in I(C), and H is locally equivalent to I(C).

K 4,4 vs. W 7
The results of the preceding section leave us with the task of distinguishing W 7 from an interlacement graph of K 4,4 . This task is a bit more difficult than one might expect. For instance, it turns out that both W 7 and an interlacement graph of K 4,4 have 42 transverse 4-circuits, 168 transverse 6-circuits and no other transverse circuit of size ≤ 7. Nevertheless, there are several ways to verify that W 7 is not an interlacement graph of K 4,4 . We detail three in this section, and mention several more at the end of the next section.
verifying this assertion, note that if a matroid has precisely two circuits then the circuit elimination property guarantees that the two circuits must be disjoint. Now, suppose a transverse matroid M of M τ (K 4,4 ) contains two disjoint circuits. As M has 8 elements and M τ (K 4,4 ) has no transverse circuit of size < 4, the two circuits must both be of size 4. If the two disjoint circuits are of the first type, then after re-indexing we may suppose they arise from the circuits 1a2b and 3c4d. Then M also contains transverse circuits of the first type corresponding to the circuits 1c2d and 3a4b. Moreover, M contains the transverse circuits of the second type corresponding to {1, 2} (or {3, 4}) and {a, b} (or {c, d}). If the two disjoint transverse circuits are of the second type, we may presume one corresponds to {1, 2} (or {3, 4}) and the other to {a, b} (or {c, d}). Then M contains the transverse circuits of the first type that correspond to the circuits 1a2b, 3c4d, 1c2d and 3a4b. Finally, it is impossible for the two disjoint circuits to include one of the first type and one of the second type.

Interlacement graphs of K 4,4 are not 3-regular
We call a simple graph strictly supercubic if all the vertices are of degree ≥ 3, and there is at least one vertex of degree > 3. If any vertex is of degree > 3 we are done, so we may suppose G is 3-regular.
Let v 1 and v 2 be nonadjacent vertices of G. If they share all their neighbors, then {χ G (v 1 ), χ G (v 2 )} is a transverse 2-circuit of M [IAS(G)]. Theorem 43 tells us that this is impossible. On the other hand, if they share no neighbor then is a transverse matroid of G that contains two disjoint circuits, and no other circuit. Again, Theorem 43 tells us that this is impossible.
According to Meringer [30], there are five simple 3-regular graphs with 8 vertices. They are displayed in Figure 11. We observe that the first of these graphs has a pair of nonadjacent vertices that share all their neighbors, and each of the next three has a pair of nonadjacent vertices that share no neighbor. It is not immediately apparent, but the fifth graph pictured in Figure 11 is locally equivalent to the fourth; see Figure 12 for details. (It is easy to see that the last graph in Figure 12 is isomorphic to the fourth graph in Figure 11, if you first observe that each has precisely two 3-circuits.) Figure 11: The simple cubic graphs of order 8. By the way, there is a considerably longer proof of Proposition 44 that may be of interest in spite of its length. Theorems 6 and 43 tell us that if G is an interlacement graph of K 4,4 then all the transverse circuits of G are of size ≥ 4, and all the vertices of G are of degree ≥ 3. Consequently, G is either cubic or strictly supercubic. According to [41], though, every 3-regular circle graph has transverse circuits of size 2; hence G is not 3-regular.
Corollary 45 Every interlacement graph of an Euler circuit of K 4,4 has a vertex of degree 5.
Proof. As observed at the beginning of this section, a computer search (again using the matroid module for Sage [33,36]) indicates that the transverse circuits of M τ (K 4,4 ) of size < 9 are all of size 4, 6 or 8. Consequently Theorem 6 guarantees that if G is an interlacement graph of an Euler circuit of K 4,4 then the vertex-degrees in G are elements of the set {3, 5, 7}. Proposition 44 assures us that G must have at least one vertex of degree 5 or 7.
Suppose G has no vertex of degree 5. If v 1 and v 2 both have degree 7 then the ψ G (v 1 ) and ψ G (v 2 ) columns of IAS(G) are the same, so {ψ G (v 1 ), ψ G (v 2 )} is a transverse 2-circuit of G. But M τ (K 4,4 ) has no transverse 2-circuit, so this cannot be the case. That is, G must have one vertex v 1 of degree 7, and seven We conclude that G must have a vertex of degree 5.

A distinctive transverse matroid of M τ (K 4,4 )
Here is another way to distinguish W 7 from an interlacement graph of K 4,4 .
Proposition 46 Let P be a 4-element circuit partition of K 4,4 . Then either T ch(P ) is obtained from a 4-cycle by doubling all edges, or T ch(P ) is obtained from a complete graph by doubling two non-incident edges. In contrast, the nullity-3 transverse matroids of W 7 are all isomorphic to the cocycle matroid of a complete graph with two non-incident edges doubled.
Proof. As K 4,4 has 16 edges and no circuit of size < 4, P must contain four 4-circuits. Let γ 1 be one of these 4-circuits, and let the vertices on γ 1 be v 1 , w 1 , v 2 , w 2 (in order). Let {v 1 , v 2 , v 3 , v 4 } and {w 1 , w 2 , w 3 , w 4 } be the two vertex classes of K 4,4 . The edges v 1 w 3 and v 1 w 4 must appear in a single circuit γ 2 of P , as they are the only remaining edges incident on v 1 . Suppose the vertices on γ 2 are v 1 , w 3 , w 4 and v 2 . Then the edges of T ch(P ) corresponding to v 1 and v 2 both connect γ 1 to γ 2 . If the two remaining circuits γ 3 , γ 4 ∈ P are incident on {v 3 , v 4 , w 1 , w 2 } and {v 3 , v 4 , w 3 , w 4 } respectively, then T ch(P ) is a 4-cycle with all edges doubled. If instead they are incident on {v 3 , v 4 , w 1 , w 3 } and {v 3 , v 4 , w 2 , w 4 }, or {v 3 , v 4 , w 1 , w 4 } and {v 3 , v 4 , w 2 , w 3 }, then T ch(P ) is a copy of K 4 with the edges γ 1 γ 2 and γ 3 γ 4 doubled.
Suppose the vertices on γ 2 are v 1 , w 3 , w 4 and v 3 . Then neither γ 1 nor γ 2 includes v 4 , so the two remaining elements of P must both be incident on v 4 . As the edges v 2 w 1 and v 2 w 2 are both included in γ 1 , one of these circuits must include the edges v 2 w 3 and v 2 w 4 . Call this one γ 3 ; then v 2 , w 3 , w 4 and v 4 all appear on γ 3 . The last element of P must be incident on {v 3 , w 1 , w 2 , v 4 }. Then T ch(P ) is a copy of K 4 with the edges γ 1 γ 4 and γ 2 γ 3 doubled.
If the vertices on γ 2 are v 1 , w 3 , w 4 and v 3 , then apply the discussion of the preceding paragraph, after interchanging the indices of v 3 and v 4 .
The assertion regarding W 7 has been verified with computer programs (again using Sage [33,36]).

Circle graph characterizations
Corollary 40 and Theorem 43 yield Theorem 10. Here is a similar characterization: Theorem 47 A simple graph G is a circle graph if and only if G satisfies the following conditions.
2. If H ∈ VM 8 (G) is of order ≤ 7, then H has a transverse circuit of size ≤ 3.
3. If H ∈ VM 8 (G) is of order 8, and the cocycle matroid of the graph of Figure 10 is a transverse matroid of H, then H has a transverse circuit of size ≤ 3.
Proof. Condition 1 guarantees that BW 3 is not a vertex-minor of G, condition 2 guarantees that W 5 is not a vertex-minor of G, and condition 3 guarantees that W 7 is not a vertex-minor of G. Consequently if G satisfies all three conditions then G is a circle graph.
On the other hand, if G is a circle graph then Theorem 36 tells us that G satisfies condition 1. To see why G satisfies conditions 2 and 3, note that if H ∈ VM 8 (G) is of order < 8, or the cocycle matroid of the graph of Figure  10 is a transverse matroid of H, then H is not an interlacement graph of K 4,4 . Corollary 41 then tells us that H has a transverse circuit of size ≤ 3.
Another similar characterization is proven in much the same way: Theorem 48 A simple graph G is a circle graph if and only if G satisfies the following conditions.
2. If H ∈ VM 8 (G) and the cocycle matroid of the graph of Figure 13 is not a transverse matroid of H, then H has a transverse circuit of size ≤ 3.
Theorem 12 follows from Corollaries 42 and 45. Proposition 46 leads to a modified version of Theorem 12: Theorem 49 A simple graph G is a circle graph if and only if every H ∈ VM 8 (G) satisfies at least one of the following conditions. 1. H is locally equivalent to a graph with a vertex of degree 0 or 1.

2.
H is locally equivalent to a graph with a pair of adjacent degree-2 vertices.
3. H is locally equivalent to a graph with three mutually nonadjacent degree-3 vertices, which have two neighbors in common.
Proof. If G is a circle graph and H ∈ VM 8 (G) does not satisfy condition 1 or condition 2, then Corollary 42 tells us that H is an interlacement graph of It follows that M is isomorphic to the cocycle matroid of the graph pictured in Figure 13. (The isomorphism is given by mapping the pairs of parallel edges in Figure 13 to the pairs {a, f }, {b, g}, {c, h}, and {d, e}.) As H has 8 vertices and M is not a transverse matroid of W 7 , H is not locally equivalent to W 5 , BW 3 or W 7 .
We conclude that VM 8 (G) does not include any graph locally equivalent to W 5 , BW 3 or W 7 . Bouchet's theorem now tells us that G is a circle graph.
Several other matroidal properties may be used to provide circle graph characterizations.
Proof. Computer programs indicate that W 7 has 42 and an interlacement graph of K 4,4 has 45.
Corollary 51 A simple graph G is a circle graph if and only if G satisfies the following conditions.
2. If H ∈ VM 8 (G) is of order ≤ 7, then H has a transverse circuit of size ≤ 3. A computer program (using Sage [33,36]) indicates that M [IAS(W 7 )] has 336 automorphisms, while M τ (K 4,4 ) has 1152 automorphisms. Also, the isotropic matroid of W 7 has an automorphism of order 7 but the transition matroid of

Bipartite circle graphs
In this section we discuss Theorem 11 of the introduction, and several associated results. These results differ from the characterizations of general circle graphs discussed above in that they do not rely on Bouchet's theorem. Their foundation is an earlier result of de Fraysseix [23]; a proof is included for the reader's convenience.
Proposition 52 [23, Proposition 6] Let G be a bipartite simple graph, with vertex classes V 1 and V 2 . Then G is a circle graph if and only if the transverse Proof. Let the adjacency matrix of G be Then the transverse matroids Suppose conversely that M 1 is a planar matroid; then M 2 is planar too. Let H be a plane graph whose cycle matroid is M 1 . We presume that H is connected, its dual H * is connected, and that H and H * may be drawn together in the plane in such a way that all their edges are smooth curves, in general position. We identify E(H) and E(H * ) with V (G).
H has a spanning tree T corresponding to the basis φ G (V 1 ) of M 1 . Choose an ε-neighborhood D of T ; D is homeomorphic to a disk, so its boundary ∂D is homeomorphic to a circle. As long as ε is sufficiently small, each element e ∈ E(H) − E(T ) intersects D in two short arcs. Use e to label the end-points of these arcs on ∂D. Also choose two points of ∂D for each edge of T , one on each side of the midpoint; label these two points with that edge. If we trace the circle ∂D and read off the labeled points in order then we obtain a double occurrence word W , in which each edge of H appears twice.
Consider an edge e ∈ E(T ), and write W as eW 1 eW 2 . Let γ be a simple closed curve obtained by following ∂D from one point labeled e to the other point labeled e, and then crossing through D back to the first point labeled e. Clearly then γ encloses one connected component of T − e. If e ′ is an edge of the connected component of T − e enclosed by γ, then both points of ∂D labeled e ′ appear on γ. If e ′ is an edge of the connected component of T − e not enclosed by γ, then neither point of ∂D labeled e ′ appears on γ. Either way, we see that e and e ′ are not interlaced with respect to W . Now consider an edge e ∈ E(H) − E(T ), and write W as eW 1 eW 2 . Suppose e ′ = e ∈ E(H) − E(T ). If e ′ appears precisely once in W 1 , then it is not possible to draw e and e ′ without a crossing outside D; as there is no such crossing, it cannot be that e ′ appears precisely once in W 1 . That is, e and e ′ are not interlaced with respect to W . Considering the preceding paragraph, we deduce that the interlacement graph I(W ) is bipartite, with E(T ) and E(H) − E(T ) as vertex-classes.
Again, let e ∈ E(H) − E(T ), and write W as eW 1 eW 2 . Remove all appearances of edges not in T from W 1 and W 2 ; the result is two walks in T connecting the end-vertices of e. If we remove the edges that appear twice in either of these walks, we must obtain the unique path in T connecting the end-vertices of e. We conclude that the fundamental circuit of e with respect to T in H coincides with the closed neighborhood of e in I(W ). As T is a spanning tree of H corresponding to the basis φ G (V 1 ) of M 1 , the fundamental circuit of e with respect to T in H is the same as the closed neighborhood of e in G.
As G and I(W ) are both bipartite simple graphs with a vertex-class corresponding to V 1 = E(T ), it follows that G = I(W ).
Note that in the situation of Proposition 52, G is a special kind of circle graph: it is an interlacement graph of a planar 4-regular graph F . To construct such an F , start with the closed curve ∂D mentioned in the proof of Proposition 52. F has a vertex at the midpoint of each edge of T , and a vertex outside D on each edge of E(H) − E(T ); edges are inserted so that the word W corresponds to an Euler circuit of F . It is a simple matter to draw F in the plane using the closed curve ∂D as a guide.
Proposition 52 implies the following.
Theorem 53 Let G be a simple graph. Then any one of the following conditions implies the rest.
1. G is the interlacement graph of an Euler system of a planar 4-regular graph.
2. There are disjoint transversals T 1 , T 2 of W (G) such that r(T 1 ) + r( is a planar matroid.
3. G is locally equivalent to a bipartite graph, and all the transverse matroids of G are cographic.
Proof. If G is a circle graph associated with a planar 4-regular graph then it is well known that G is locally equivalent to a bipartite circle graph; see [34] for instance. This and Theorem 36 give us the implication 1 ⇒ 3. Suppose condition 3 holds, and let H be a bipartite graph locally equivalent to G. As there is an induced isomorphism between the isotropic matroids of G and H, we may verify condition 2 for G by verifying it for H. Let V (H) = V 1 ∪V 2 with V 1 and V 2 both stable sets of H. Then the adjacency matrix of H is Consider the transversals is bipartite with vertex-classes β(V 1 ) and β(V 2 ). It follows from (b) that for . Proposition 52 and the subsequent discussion imply that condition 1 holds. It is important to realize that although BW 3 is the only one of Bouchet's circle graph obstructions with non-cographic transverse matroids, condition 3 of Theorem 53 does not imply that a bipartite non-circle graph must have BW 3 as a vertex-minor. For instance, a computer search using Sage [33,36] indicates that BW 3 is not a vertex-minor of the bipartite graph BW 4 pictured on the left in Figure 14. Nevertheless BW 4 is not a circle graph. One way to verify this assertion is to observe that the transverse matroid {χ 1 , φ 2 , χ 3 , φ 4 , χ 5 , φ 6 , χ 7 , φ 8 , φ 9 } is not cographic. Indeed, this transverse matroid is isomorphic to the cycle matroid M (K 3,3 ); an isomorphism is indicated by the labels 1, 2, ..., 9 in the figure. (Notice that the labels denote vertices of BW 4 and edges of K 3,3 .) Another way to verify that BW 4 is not a circle graph is to obtain W 5 as a vertex-minor; this can be done by performing local complementations with respect to vertices 2, 4 and 7, and then removing them.
Theorem 11 of the introduction follows from Theorem 53 and one more result: Corollary 54 Let G be a bipartite simple graph. Then G is a circle graph if and only if neither BW 3 nor BW 4 is a vertex-minor.
Proof. Of course, if BW 3 or BW 4 is a vertex-minor then G is not a circle graph.
For the converse, suppose G is not a circle graph. If V 1 and V 2 are the vertex-classes of G then Proposition 52 tells us that the transverse matroid M = φ G (V 1 ) ∪ χ G (V 2 ) is not a planar matroid. As noted in the proof of Proposition 52, the transverse matroid M * = φ G (V 2 ) ∪ χ G (V 1 ) is the dual of M ; so of course M * is not planar either.
Suppose for the moment that M is minor-minimal with regard to nonplanarity. Then M or M * is isomorphic to one of F 7 , M (K 3,3 ), M (K 5 ), so G is a fundamental graph of one of these matroids. The fundamental graphs of a binary matroid are all equivalent under edge pivots, so they are certainly locally equivalent; hence it suffices to verify that one fundamental graph of each of F 7 , M (K 3,3 ), and M (K 5 ) has BW 3 or BW 4 as a vertex-minor. BW 3 is a fundamental graph of F 7 , and BW 4 is a fundamental graph of M (K 3,3 ). For K 5 , consider the fundamental graph pictured on the left in Figure 15. (The labels indicate an isomorphism between the transverse matroid {φ 1 , χ 2 , χ 3 , χ 4 , χ 5 , χ 6 , χ 7 , φ 8 , φ 9 , φ 10 } and M (K 5 ).) Clearly BW 3 is a vertex-minor, obtained by first performing local complementations at the vertices 2, 3, 4, and then removing them.
Proceeding inductively, suppose that M is not minor-minimal with regard to nonplanarity. Then there is an m ∈ M such that M/m or M − m is also nonplanar. If M − m is nonplanar, then M * /m = (M − m) * is nonplanar. Consequently by interchanging V 1 and V 2 if necessary, we may presume that M/m is nonplanar.
If m = φ G (v) ∈ φ G (V 1 ) then M/m is a transverse matroid of G − v , so applying the inductive hypothesis to G−v implies that BW 3 or BW 4 is a vertexminor of G. If m = χ G (v) ∈ χ G (V 2 ) is a loop of M , then v is an isolated vertex of G, and again we may apply the inductive hypothesis to G−v. If m = χ G (v) ∈ χ G (V 2 ) is not a loop of M , then v has a neighbor w ∈ V 1 . In this case the edge pivot G vw is a bipartite graph with vertex classes V 1 ∆{v, w} and V 2 ∆{v, w}, and an induced isomorphism β : M [IAS(G)] → M [IAS(G vw )] has β(φ G (w)) = χ G vw (w) and β(χ G (v)) = φ G vw (v) [39]. As β(M ) is isomorphic to M , it too is nonplanar; and so is β(M )/φ G vw (v) ∼ = M/χ G vw (v). As β(M )/φ G vw (v) is a transverse matroid of G vw − v, the inductive hypothesis implies that BW 3 or BW 4 is a vertex-minor of G vw , and hence of G.

Crossing numbers of 4-regular graphs
A simple construction indicates that all 4-regular graphs can be obtained from planar 4-regular graphs. Examples of the construction appear in the second row of Figure 9.
Theorem 55 Every 4-regular graph is a detachment of a planar 4-regular graph.
Proof. Draw a 4-regular graph F in the plane, with its edges in general position. That is, the only failures of planarity are points where two edges cross. To obtain a planar graph with F as a detachment, replace each edge-crossing with a vertex.
This leads to yet another characterization of circle graphs: Corollary 56 Every circle graph is a vertex-minor of a bipartite circle graph.
Consequently, a simple graph is a circle-graph if and only if it is a vertexminor of some graph that satisfies Theorem 53. Considering condition 2 of Theorem 53 in particular, one might hope that Corollary 56 would lead to a characterization of circle graphs using some matroidal property of the unions of pairs of transversals. We do not know what such a property might be, though.
Proposition 57 Let G be a simple graph. If G is the interlacement graph of an Euler system of a 4-regular graph of crossing number k, then there are disjoint transversals T 1 , T 2 of W (G) such that r(T 1 ) + r(T 2 ) ≤ |V (G)| + k.
Proof. If k = 0 then the result follows from Theorem 53. Proceeding inductively, suppose k ≥ 1 and F is a 4-regular graph of crossing number k with G as interlacement graph. Let F ′ be the graph obtained by replacing one crossing with a vertex v. Then by the inductive hypothesis M τ (F ′ ) has disjoint transversals T ′ 1 , T ′ 2 such that r(T ′ 1 ) + r(T ′ 2 ) ≤ |V (G)| + 1 + k − 1 = |V (G)| + k. As M [IAS(G)] = M τ (F ) is an isotropic minor of M τ (F ′ ) obtained by removing the vertex triple of v, T ′ 1 and T ′ 2 yield disjoint transversals T 1 , T 2 of W (G). Contraction and deletion certainly cannot raise ranks, so r(T 1 ) ≤ r(T ′ 1 ) and r(T 2 ) ≤ r(T ′ 2 ). Proposition 57 detects the crossing numbers of some 4-regular graphs. For instance, consider the 4-regular graph F pictured in the lower left-hand corner of Figure 9. Observe that F has precisely four 3-circuits, in two pairs each of which has a shared edge. No circuit partition can include two 3-circuits that share an edge, so a circuit partition of F includes at most two 3-circuits. As F has 16 edges, it follows that every circuit partition includes ≤ 4 circuits. Theorem 36 then tells us that every transversal of M τ (F ) is of rank ≥ 5, so the smallest possible value of r(T 1 ) + r(T 2 ) is 10. According to Proposition 57, this fact guarantees that the crossing number of F is ≥ 2; as the drawing in Figure  9 has two crossings, we conclude that the crossing number of F is 2.
Proposition 57 is not always so precise, though. For instance, the reader will have little trouble finding a pair of 4-element circuit partitions in K 4,4 that do not share any transition. The corresponding transversals of M τ (K 4,4 ) are of rank 5, so they satisfy Proposition 57 with k = 2. But it is well known that the crossing number of K 4,4 is 4, not 2.
Let F now denote the 7-vertex 4-regular graph pictured in the middle of the top row of Figure 9. The figure suggests that the crossing number of F is 2, and it is not very hard to show that this is indeed the case. However Figure 16  indicates two partitions of E(F ) into four circuits. According to Theorem 36 both of the corresponding transversals of M τ (F ) have rank 4, so they satisfy the necessary condition of Proposition 57 with k = 1. They do not satisfy the stronger necessary condition of Proposition 58, though.
Proposition 58 Let G be a simple graph. If G is the interlacement graph of an Euler system of a 4-regular graph of crossing number 1, but G is not the interlacement graph of an Euler system of any planar 4-regular graph, then there are disjoint transversals T 1 , T 2 of W (G) such that r(T 1 ) + r(T 2 ) = |V (G)| + 1 and M [IAS(G)] | (T 1 ∪ T 2 ) is a planar matroid. It remains only to note that Theorem 53 tells us that the inequality r(T 1 ) + r(T 2 ) ≤ |V (F )| + 1 must be an equality, for otherwise G would be the interlacement graph of an Euler system of a planar 4-regular graph.
We do not know whether it is possible to significantly sharpen Proposition 57 for k > 1. If it is possible, examples indicate that the sharpened version must be quite different from Proposition 58. For the graph F pictured in Figure 16, a computer search using Sage [33,36] finds that there are no two disjoint transversals of M τ (F ) whose union is a planar matroid. The situation in M τ (K 4,4 ) is even more restrictive: there are no two disjoint transversals whose union is a regular matroid.