A Characterization of Circle Graphs in Terms of Multimatroid Representations

The isotropic matroid M [IAS(G)] of a looped simple graph G is a binary matroid equivalent to the isotropic system of G. In general, M [IAS(G)] is not regular, so it cannot be represented over fields of characteristic 6= 2. The ground set of M [IAS(G)] is denoted W (G); it is partitioned into 3-element subsets corresponding to the vertices of G. When the rank function of M [IAS(G)] is restricted to subtransversals of this partition, the resulting structure is a multimatroid denoted Z3(G). In this paper we prove that G is a circle graph if and only if for every field F, there is an F-representable matroid with ground set W (G), which defines Z3(G) by restriction. We connect this characterization with several other circle graph characterizations that have appeared in the literature. Mathematics Subject Classifications: 05C31


Introduction
In this paper a graph may have loops or parallel edges. A graph is simple if it has neither loops nor parallels, and a looped simple graph has no parallels. An edge consists of two distinct half-edges, each incident on one vertex; and an edge is directed by distinguishing one of its half-edges as initial. The degree of a vertex is the number of incident half-edges, and a graph is d-regular if its vertices are all of degree d. We use the term circuit to refer to a sequence v 1 , h 1 , h 2 , v 2 , h 2 , . . . , h k−1 , h k = h 1 , v k = v 1 of vertices and half-edges, such that: for each i < k, h i and h i are half-edges incident on v i ; for each i < k, h i and h i+1 are half-edges of the same edge; and the half-edges h 1 , h 1 , . . . , h k−1 , h k−1 are pairwise distinct. We consider two circuits to be the same if they differ only in orientation or starting point.
If G is a connected graph whose vertices are all of even degree then G has an Euler circuit, i.e., a circuit which includes every edge of G. In general, if the vertices of G are It follows from [27,Proposition 41] that the isotropic 3-matroid of a graph is a multimatroid, a notion introduced by Bouchet [4,5,6,7]. Like ordinary matroids, multimatroids are uniquely determined by their bases, circuits, or independent sets. An independent set of Z 3 (G) is a subtransversal I ∈ S(G) with r(I) = |I|. A circuit of Z 3 (G) is a subtransversal C ∈ S(G) that is minimal (w.r.t. inclusion) with the property that it is not an independent set. A basis of Z 3 (G) is an independent set of Z 3 (G) that is maximal (w.r.t. inclusion) with this property. It follows from [4,Proposition 5.5] that all bases of Z 3 (G) have a common cardinality equal to |Ω| = |V (G)|. Consequently, if X is the set of bases (independent sets, circuits, resp.) of M [IAS(G)], then X ∩ S(G) is the set of bases (independent sets, circuits, resp.) of Z 3 (G).
Definition 5. Let F be a field. Then Z 3 (G) is representable over F if there is an F-matrix B with columns indexed by W (G), such that the rank function of Z 3 (G) agrees with the rank function of B when restricted to S(G).
To say the same thing in a different way: Z 3 (G) is representable over F if and only if there is an F-representable matroid M on W (G), whose rank function agrees with the rank function of M [IAS(G)] when restricted to S(G). Such a matroid M is said to shelter Z 3 (G). Notice that Z 3 (G) is GF (2)-representable by definition; it is sheltered by M [IAS(G)]. Our main result is that representability over other fields characterizes circle graphs. Theorem 6. Let G be a simple graph. Then any one of the following conditions implies the others.
1. G is a circle graph.
Theorem 6 shows that the theory of isotropic 3-matroids is quite different from the more familiar theory of graphic matroids: all graphic matroids are representable over all fields, but the only isotropic 3-matroids representable over all fields are those that come from circle graphs.
Here is an outline of the paper. In Section 2 we provide some details about sheltering matroids for Z 3 (G). In Section 3 we deduce the implication 3 ⇒ 1 of Theorem 6 from the results of Section 2 and Bouchet's circle graph obstructions theorem [3]. In Section 4 we summarize the signed interlacement machinery of [30], which associates matrices over GF (2) and R with circuit partitions in 4-regular graphs. In Section 5 we use this machinery to show that if G is a circle graph, then Z 3 (G) is representable over R. This argument is fairly direct, and suffices to prove the implication 1 ⇒ 3 of Theorem 6. Getting Condition 2 into the picture is more difficult, because the matrix machinery of [30] fails over fields with char(F) > 2. In Section 6 we develop a special case of the signed interlacement machinery, which works over all fields. In Section 7 we complete the proof of Theorem 6, and in Section 8 we detail the constructions used in the proof for a small example. In Sections 9 and 10 we discuss the connections between Theorem 6 and other characterizations of circle graphs that have appeared in the literature.

Representations of sheltering matroids
We begin by recalling the definition of local equivalence.
Definition 7. Let G be a looped simple graph and v a vertex of G.
• The graph obtained from G by complementing the loop status of v is denoted G v .
• The graph obtained from G by complementing the adjacency status of every pair of neighbors of v is the simple local complement of G at v, denoted G v s .
• The graph obtained from G by complementing the adjacency status of every pair of neighbors of v and the loop status of every neighbor of G is the non-simple local complement of G at v, denoted G v ns .
• The equivalence relation on looped simple graphs generated by loop complementations and local complementations is local equivalence.
For a graph G and X ⊆ V (X), we denote the subgraph of G induced by V (G) \ X by G − X.
Definition 8. Let G and H be looped simple graphs. Then H is a vertex-minor of G if there is a graph G that is locally equivalent to G, such that H = G − X for some subset X ⊆ V (G ).
In analogy with Definition 3, if G is a looped simple graph with adjacency matrix A(G), then we define the restricted isotropic matroid M [IA(G)] to be the matroid represented by the matrix IA(G) = I A(G) over GF (2). That is, M [IA(G)] is the submatroid of M [IAS(G)] that includes only φ and χ elements. The isotropic 2-matroid Z 2 (G) is the 3-tuple (U, Ω, r), where Ω = {{φ G (v), χ G (v)} | v ∈ V (G)}, U = Ω, and r is the restriction of the rank function of M [IA(G)] to subtransversals S ∈ S(G) that involve only φ and χ elements. A sheltering matroid for Z 2 (G) is a matroid with the same ground set U , whose rank function restricts to the rank function of Z 2 (G). If F is a field and Z 2 (G) has an F-representable sheltering matroid, then Z 2 (G) is F-representable.
If M is a sheltering matroid for Z 2 (G), then this transversal must be dependent in M too. Hence the φ G elements span M , so the rank r(M ) is no more than the number of φ G elements, which is |V (G)|. On the other hand, Proof. If M is a sheltering matroid for Z 3 (G) then the submatroid N consisting of elements of M that correspond to elements of Z 2 (G) is a sheltering matroid. If M is a strict sheltering matroid then is also a basis of N ; hence r(N ) = |V (G)|.

1.
A has nonzero entries in precisely the same places where the adjacency matrix A(G) has nonzero entries.
2. I A , with I an identity matrix, represents a matroid that shelters Z 2 (G), with the columns of I and A corresponding to the φ G and χ G elements, respectively.
Proof. As Z 2 (G) is F-representable, for some m there is an m × 2n matrix Q with entries in F, which represents a matroid that shelters Z 2 (G). We presume the columns of Q are ordered with the φ columns first, and then the χ columns. The φ G elements of M [IAS(G)] are independent, so the first n columns of Q are linearly independent. It follows that elementary row operations can be used to bring Q into the form Elementary row operations have no effect on the matroid represented by a matrix, so Q represents a matroid that shelters Z 2 (G). If any entry of A is not 0, then the corresponding column of Q is not included in the span of the columns of I. Consequently if v ∈ V (G) is the vertex whose χ G (v) element corresponds to this column of Q , then {χ G (v)} ∪ {φ G (w) | w = v} is an independent set of the matroid represented by Q . This set is a transversal of W (G), so it is also an independent set of M [IAS(G)]. This is incorrect, however; if and only if v and w are neighbors in G. By hypothesis, this subtransversal is independent in M [IAS(G)] if and only the corresponding columns of Q are linearly independent. As the columns of Q corresponding to φ G elements are columns of the identity matrix, and the w entry of the χ G (w) column of Q is 0, it follows that the v entry of this column of Q is 0 if and only if the v entry of the corresponding column of IAS(G) is 0.
Lemma 14. Let G be a simple graph with n vertices, such that Z 3 (G) is F-representable. Then for some m ≥ n, there is an m × 3n matrix P = I A B 0 0 C with entries in F, which satisfies the following.
1. P represents a matroid that shelters Z 3 (G), with the columns of I, A and B corresponding to the φ G , χ G and ψ G elements, respectively.
2. The submatrix I is an n × n identity matrix.
3. The submatrix A has nonzero entries in precisely the same places where the adjacency matrix A(G) has nonzero entries.
If Z 3 (G) is strictly F-representable, then there is a matrix P which moreover satisfies the three conditions below.
5. The diagonal entries of the submatrix B are all equal to 1.

6.
If v and w are neighbors in G, then B vw B wv = 1.
Proof. Let Q be an m × 3n matrix with entries in F, which represents a matroid that shelters Z 3 (G). We presume the columns of Q are ordered with the φ columns first, then the χ columns, and then the ψ columns. The φ G elements of M [IAS(G)] are independent, so the first n columns of Q are linearly independent. Elementary row operations can be used to bring Q into the form Elementary row operations have no effect on the matroid represented by a matrix, so P inherits property 1 from Q.
The proof of Lemma 13 shows that A = 0 and A satisfies property 3. If Z 3 (G) is strictly F-representable, we may start with a matrix Q of rank n. Then P also is of rank n, so A = C = 0. If . It follows that the corresponding columns of P are linearly independent; this requires that B vv = 0. We may multiply the v column of B by 1/B vv , without affecting the represented matroid. For simplicity we still use B and P to denote the matrices resulting from these column multiplications.
It remains to verify property 6. If v and w are neighbors in G then It follows that the corresponding columns of P are linearly dependent, so Notice that property 3 of Lemma 14 specifies the locations of nonzero entries in the submatrix A. We do not have such specific information about B, however. Properties 5 and 6 guarantee nonzero entries on the diagonal, and in locations that correspond to edges of G; but there may be nonzero entries in other places, and these locations may vary from one sheltering matroid to another.

Bouchet's obstructions
In this section we prove the implication 3 ⇒ 1 of Theorem 6: if Z 3 (G) is F-representable over some field with char(F) = 2, then G is a circle graph. The crucial ingredient of the proof is the following well-known result.
Theorem 15. (Bouchet's circle graph obstructions theorem [3]) A simple graph is not a circle graph if and only if it has one of the graphs pictured in Figure 1 as a vertex-minor.
Bouchet's theorem is useful in proving Theorem 6 because of the following result. This result is closely related to a statement given (without proof) in [16, page 36] in the context of delta-matroids.
Proposition 16. If F is a field with char(F) = 2 and G ∈ {W 5 , BW 3 , W 7 }, then Z 2 (G) is not representable over F.
Proof. There is a transversal T of W (BW 3 ) which includes only φ and χ elements, such that the restriction of M [IAS(BW 3 )] to T is isomorphic to the Fano matroid F 7 . The Fano matroid is not representable over F, so no F-representable matroid can shelter Z 2 (BW 3 ).
The proposition is a bit harder to verify for W 5 . To establish notation, we set V (W 5 ) = {1, 2, 3, 4, 5, 6} and E( If we multiply a column of A by a nonzero element of F, then we do not change the matroid represented by I A . Consequently we may presume that in each column of A, the first nonzero entry is 1. Property 1 of Lemma 13 now tells us that where the entries represented by letters are nonzero elements of F. The submatrix of IAS(W 5 ) corresponding to the subtransversal {φ W 5 (4), φ W 5 (5), χ W 5 (3), χ W 5 (6)} is represented by the rank 3 matrix is also of rank 3. We deduce that b = b . Similar arguments tell us that d = d , f = f , h = h and j = j .
A direct calculation shows that the determinant of this matrix is 2bdf hj, which is nonzero in F but 0 in GF (2). It follows that the transversal T is independent in M [ I A ] and dependent in M [IAS(G)]; this contradicts property 2 of Lemma 13.
The proposition may be verified for W 7 by a closely analogous argument.
We deduce the contrapositive of the implication 3 ⇒ 1 of Theorem 6.
Corollary 17. If Z 3 (G) is representable over some field F with char(F) = 2, then G is a circle graph.
Proof. Suppose Z 3 (G) is representable over a field F with char(F) = 2. Lemmas 10 and 12 tell us that for every vertex-minor H of G, Z 2 (H) is also representable over F. According to Proposition 16, it follows that no vertex-minor of G is isomorphic to W 5 , BW 3 or W 7 . Theorem 15 now tells us that G is a circle graph.
Before proceeding we take a moment to observe that in general, Lemma 12 and Corollary 17 do not hold for Z 2 (G). Let G be a bipartite graph which is a fundamental graph for the cycle or cocycle matroid of some nonplanar graph. Then M [IA(G)] is isomorphic to the direct sum M ⊕ M * , where M is the cycle matroid of the nonplanar graph. It follows that M ⊕ M * is a strict sheltering matroid for Z 2 (G). It is well known that graphic and cographic matroids are representable over all fields [23, Lemma 5.1.3 and Corollary 5.1.6], so M ⊕ M * is representable over all fields. Hence Z 2 (G) is strictly representable over all fields. However a theorem of deFraysseix [14] asserts that G cannot be a circle graph.
Two such examples were discussed in [11,Section 8]. They are pictured in Figure 2; G 1 is a fundamental graph for M (K 5 ) and G 2 is a fundamental graph for M (K 3,3 ). It is not hard to directly confirm deFraysseix's assertion that neither G 1 nor G 2 is a circle graph [14]; G 1 has BW 3 as a vertex-minor, and G 2 has W 5 as a vertex-minor.

Circuit partitions in 4-regular graphs
In this section we summarize some ideas and results from [30]; we refer to that paper for a more detailed discussion.
If F is a 4-regular graph, then, at each vertex v ∈ V (F ), there are three different transitions, i.e., partitions of the four incident half-edges into two pairs. We use T(F ) to denote the set of transitions in F , and we refer to one pair of half-edges incident at v as a single transition. If C is an Euler system of F , then C can be used to label the elements of T(F ) in the following way. First, orient each circuit of C. Then the transition at v that pairs together half-edges which appear consecutively on the incident circuit of C is denoted φ C (v); the other transition that is consistent with the edge directions defined by the incident circuit of C is denoted χ C (v); and the third transition, which is inconsistent with the edge directions defined by the incident circuit of C, is denoted ψ C (v). These transition labels are not changed if the orientations of some circuits of C are reversed.
The reappearance of the φ, χ, ψ symbols used to label elements of isotropic matroids is no coincidence. If C is an Euler system of F , then there is a bijection W (I(C)) ↔ T(F ) given by This bijection relates each transversal T ∈ T (I(C)) to a partition of E(F ) into edgedisjoint circuits, and it turns out that the 3-matroid Z 3 (I(C)) is determined by these partitions. Before giving details we should emphasize that according to the definition given in the introduction, for us a "circuit" is a closed trail. In particular a circuit in a 4-regular graph may not visit any half-edge more than once, but it may visit a vertex twice.
Definition 18. Let F be a 4-regular graph. A circuit partition of F is a partition of E(F ) into edge-disjoint circuits.
A circuit partition P is determined by choosing one transition P (v) at each vertex v of F . There are three transitions at each vertex, so there are 3 |V (F )| circuit partitions. Definition 19. Let P be a circuit partition in a 4-regular graph F . Then the touch-graph T ch(P ) is the graph with a vertex v γ for each γ ∈ P and an edge e v for each v ∈ V (F ), such that e v is incident on v γ in T ch(P ) if and only if γ is incident on v in F . Figure 3. On the left we see two circuit partitions P and P in a 4-regular graph F . To follow a circuit of P or P walk along the edges of F , making sure to maintain the line status (plain, heavy, or dashed) when traversing a vertex. (The line status may change in the middle of an edge.) On the right we see T ch(P ) and T ch(P ).

Examples of Definition 19 appear in
If P is a circuit partition in F , then every half-edge h in F has a "shadow" half-edge π P (h) in T ch(P ), defined in the following way. If h is incident on a vertex v, γ is the circuit of P that includes h, and {h, h } is the single transition of P that includes h, then π P (h) = π P (h ) is a half-edge of T ch(P ) that is contained in e v and incident on v γ . For simplicity we use the notation π P (h) = h when P is clear from the context. Also, every walk W = v 1 , e 1 , v 2 , . . . , e k−1 , v k in F has a "shadow" π P (W ) = W , which is a walk in T ch(P ). If γ 1 is the circuit of P that includes e 1 , then v γ 1 is the first vertex of W . As we follow W in F , each time we pass through a vertex v i we traverse two half-edges incident on v i ; say h before v i , and h after v i . If the transition determined by {h, h } is not a transition of P , then the edge {h, h } of T ch(P ) is added to W . If the transition determined by {h, h } is a transition of P , then no edge is added to W . (In this instance we are walking along a circuit of P as we pass through v on W , so the "shadow" is standing still on a vertex of T ch(P ).) If W is a closed walk in F then W is a closed walk in T ch(P ). Of course W may be much shorter than W ; for instance if W ∈ P then W is just v W .
Let D be a directed version of T ch(P ). Let F be a field, and for each directed walk W in T ch(P ) let z D (W ) be the vector in F E(T ch(P )) obtained by tallying +1 in the e coordinate each time W traverses e in accordance with the D direction, and −1 in the e coordinate each time W traverses e against the D direction. Then the subspace of F E(T ch(P )) spanned by {z D (W ) | W is a closed walk in T ch(P )} is the cycle space of T ch(P ) over F, denoted Z D (T ch(P )). We refer to standard texts in graph theory, like [1], for detailed discussions of cycle spaces.
Definition 20. Let F be a 4-regular graph with an Euler system C. For each v ∈ V (F ), there are two circuits obtained by following a circuit of C from v to v. These are the fundamental circuits of C at v.
Theorem 21. ([30, Section 2]) Let C be an Euler system of a 4-regular graph F , and let Γ be a set that includes one fundamental circuit of C at each v ∈ V (F ), along with an orientation for that circuit. Then for every circuit partition P of F and every directed version We think of Theorem 21 as a surprising result because the touch-graphs of circuit partitions in F are quite varied. There are touch-graphs in which all edges are loops (the touch-graphs of the smallest circuit partitions, the Euler systems), touch-graphs in which no edges are loops (the touch-graphs of the maximal circuit partitions), and many other touch-graphs between these extremes. Despite this variation, Theorem 21 allows us to describe spanning sets in the cycle spaces of all touch-graphs in a consistent way.
When citing Theorem 21, we use the notation Then the cycle space Z D (T ch(P )) over R is the row space of the V (F )×V (F ) matrix M R,Γ (C, P, D) whose v row is the vector z D (C Γ (v)). Over GF (2), Z D (T ch(P )) is the row space of the matrix M (C, P ) obtained from M R,Γ (C, P, D) by reducing all entries modulo 2. We can use the simple M (C, P ) notation when we work over GF (2) because the value of M R,Γ (C, P, D) vw modulo 2 is independent of both D and Γ.
Theorem 21 implies that the matroid defined by M R,Γ (C, P, D) over R is the same as the matroid defined by M (C, P ) over GF (2): consisting of columns corresponding to elements of S, and let M (S) be the matrix obtained from M R (S) by reducing its entries modulo 2. Then the rank of M (S) over GF (2) is the same as the rank of M R (S) over R.
Proof. Suppose that S is minimal among subsets of V (F ) for which the rank of M (S) over GF (2) is not the same as the rank of M R (S) over R. If the rank of M (S) over GF (2) is strictly larger than the rank of M R (S) over R, then the columns of M (S) are linearly independent over GF (2) but the columns of M R (S) are linearly dependent over R, and hence also over Q. That is, there are rational numbers r s , s ∈ S, not all 0, such that if we multiply the s column of M R (S) by r s for each s ∈ S, then the sum of the resulting column vectors is 0. Eliminating common factors and multiplying by denominators, we may presume that the numbers r s are relatively prime integers. Then not all of the r s are divisible by 2, so they define a linear dependence of the columns of M (S) over GF (2), contradicting the hypothesis that the columns of M (S) are independent over GF (2). We conclude that the rank of M (S) over GF (2) is strictly smaller than the rank of M R (S) over R.
Thus the minimality of S guarantees that the columns of M R (S) are linearly independent over R, but linearly dependent when their entries are reduced modulo 2, and no proper subset of the columns of M (S) is dependent. Let κ(S) ∈ GF (2) V (F ) be the vector whose v entry is 1 if and only if v ∈ S. Then κ(S) is an element of the orthogonal complement of the row space of M (C, P ) over GF (2), so Theorem 21 tells us that {e s | s ∈ S} is a cocycle of T ch(P ). That is, there is a proper subset P 0 of P such that S is the set of vertices of F incident on both a circuit from P 0 and a circuit from is an element of the orthogonal complement of Z D (T ch(P )) over R. But according to Theorem 21, this contradicts the hypothesis that the columns of M R (S) are linearly independent over R.
Notice that if Γ is changed by reversing the orientation of C Γ (v), then the v row of M R,Γ (C, P, D) is multiplied by −1. Also, if D is changed by reversing the direction of e w , then the w column of M R,Γ (C, P, D) is multiplied by −1.
The purpose of the subindex R in M R,Γ (C, P, D) is to remind us that in general, Theorem 21 fails over fields F with char(F) > 2. As an example, consider the 4regular graph F with V (F ) = {a, b, c}, which has two edges connecting each pair of vertices. We index the edges of F as e 1 , . . . , e 6 in such a way that an Euler circuit C is a, e 1 , b, e 2 , c, e 3 , a, e 4 , b, e 5 , c, e 6 . Let P be the circuit partition given by the transitions χ C (a), χ C (b), and ψ C (c). Then P has only one element, the Euler circuit a, e 1 , b, e 5 , c, e 2 , b, e 4 , a, e 6 , c, e 3 . Choose a directed version D of T ch(P ) so that the initial half-edge of e a involves the single transition {e 1 , e 3 }, the initial half-edge of e b involves the single transition {e 1 , e 5 } and the initial half-edge of e c involves the single transition {e 2 , e 5 }. If Γ = {e 1 e 2 e 3 , e 5 e 6 e 1 , e 3 e 4 e 5 } then det M R,Γ (C, P, D) = det A weak version of Theorem 21 does hold over all fields: Proposition 23. Let C be an Euler system of a 4-regular graph F , let Γ be a set of oriented fundamental circuits of C, and let F be a field. Then for every circuit partition P of F and every directed version Proof. If γ ∈ Γ, then γ is a closed walk in T ch(P ), so z D (γ) is an element of Z D (T ch(P )).

Theorem 6 over R
In this section we show that if F is a 4-regular graph with an Euler system C, then the machinery of Section 4 provides an R-representable sheltering matroid for Z 3 (I(C)). The basic idea is to construct a single matrix which contains M R,Γ (C, P, D) matrices for all circuit partitions of F . In order to do this we need a systematic way to choose oriented versions of touch-graphs. The approach we use is not the only possible one, but it is convenient because it is easy to describe and it is connected with signed interlacement systems that have been discussed by several authors [2,18,19,20,29,30]. Let C be an oriented Euler system of a 4-regular graph F , i.e., each circuit of C is given with an orientation. Let Γ o be a set of consistently oriented fundamental circuits of C, i.e., each C Γ o (v) ∈ Γ o is oriented consistently with the circuit of C that contains it. For each v ∈ V (F ) we designate one passage of C through v as v − and the other passage of C through v as v + , in such a way that when we follow a circuit of C in accord with its given orientation, we have . .

Definition 24.
We use these vertex signs to define an integer matrix as follows.
The columns of IAS Γ o (C) are indexed by elements of T(F ) in the following way: for each w ∈ V (F ) the w column of I corresponds to φ C (w), the w column of A corresponds to χ C (w), and the w column of B corresponds to ψ C (w).
Definition 25. Let C be an oriented Euler system of F , Γ o a set of consistently oriented fundamental circuits of C, and P a circuit partition of F .
Proposition 26. Let C be an Euler system of F , Γ o a set of consistently oriented fundamental circuits of C, and P a circuit partition of F . Then the submatrix of We see that the w column of the identity matrix is the same as the w column of M R,Γ o (C, P, D Γ o ).
If P (w) = χ C (w) then h 3 , w, h 2 and h 1 , w, h 4 are the passages of circuit(s) of P through w. The initial half-edge of e w in D Γ o is h 3 = h 2 , and the terminal half-edge is If P (w) = ψ C (w) then h 1 , w, h 3 and h 2 , w, h 4 are the passages of circuit(s) of P through w. The initial half-edge of e w in D Γ o is h 3 = h 1 , and the terminal half-edge is  Proof. If T is a transversal of W (G) then F has a circuit partition P , determined by the transitions corresponding to elements of T . The submatrix of IAS Γ o (C) corresponding to T is formed by using the columns of IAS Γ o (C) corresponding to these transitions, so Proposition 26 tells us that this submatrix is M R,Γ o (C, P, D Γ o ). Corollary 22 tells us that the rank of each set S ⊆ T of columns of this matrix is the same over GF (2) and R.
It is clear from Definition 24 that when we reduce IAS Γ o (C) modulo 2, the resulting matrix is IAS(I(C)); hence the last sentence of the preceding paragraph tells us that M R [IAS Γ o (C)] is a 3-sheltering matroid for Z 3 (I(C)). It is also clear from Definition 24 that the rank of

Based sets of fundamental circuits
Let F be a 4-regular graph, with an Euler system C and an edge e. Definition 20 implies that for each vertex v in the same connected component of F as e, one fundamental circuit of C at v includes e, and the other excludes e.
Definition 28. Let E be a set that contains one edge from each component of F , and C an Euler system of F . A set of oriented fundamental circuits of C that exclude every element of E is said to be based at E. If Γ is based at E then we also say that the elements of E are base edges for Γ.
We use the notation Γ E to indicate that Γ is based at E. Notice that if we are given C and E then there are 2 |E| different sets denoted Γ E , distinguished by the orientations of the fundamental circuits.
The usefulness of based fundamental circuits stems from the following.
Theorem 29. Let E be a fixed set of base edges in a 4-regular graph F , let C and C be Euler systems of F , and let Γ E and Γ E be based sets of oriented fundamental circuits of C and C (respectively). Then these two properties hold.
I. For every circuit partition P and every directed version D of T ch(P ), the matrices M R,Γ E (C, P, D) and M R, Γ E ( C, P, D) are row equivalent over Z. That is, one matrix can be obtained from the other using the operations "multiply a row by −1" and "add an integer multiple of one row to a different row".
II. There is a single sequence of row operations which transforms M R,Γ E (C, P, D) into M R, Γ E ( C, P, D) for every choice of P and D.
Proof. By Kotzig's theorem, it suffices to verify properties I and II when C and C are related through a single κ-transformation, C = C * v. Property I. Suppose first that v is a vertex with P (v) = ψ C (v). Let C be the Euler system of F obtained from C by reversing the direction in which the circuit of C incident at v traverses C Γ E (v). It is easy to see that this partial reversal affects some transition labels at vertices that appear precisely once on , and if w = v appears precisely once on C Γ E (v) then χ C (w) = ψ C (w) and ψ C (w) = χ C (w). Otherwise, transition labels with respect to C and C are the same.
In order to avoid proliferation of subcases we assume that the Euler circuits included in C are given with orientations, the circuits of C inherit orientations from the circuits of C, and all the fundamental circuits included in Γ E and Γ E respect these orientations. As mentioned before the example at the end of the previous section, this presumption does not involve a significant loss of generality because the effect of reversing the orientation of a fundamental circuit is simply to multiply the corresponding row of M R,Γ E (C, P, D) or M R, Γ E ( C, P, D) by −1. 1 . Then C Γ E (w) and C Γ E (w) are the same circuit in F , so they provide the same w row for M R,Γ E (C, P, D) and (It may be that v = w.) Then C Γ E (w) and C Γ E (w) are the same circuit in F , but oriented in opposite directions. It follows that the w row of M R,Γ E (C, P, D) is the negative of the w row of M R, Γ E ( C, P, D).
3. Suppose w = v ∈ V (F ) and and C Γ E (w) contains C Γ E (v). Then the vectors z D (C Γ E (w)) and z D ( C Γ E (w)) are obtained using the same contributions from passages through vertices outside C Γ E (v), and opposite contributions from passages through vertices inside C Γ E (v), other than v itself. It follows that if x = v ∈ V (F ) then the x coordinates of z D ( C Γ E (w)) and z D (C Γ E (w)) − 2 · z D (C Γ E (v)) are the same. In contrast, the situation at v is complicated by the fact that the passages of C Γ E (v) and C Γ E (w) through v are different. We index the half-edges incident on v as h 1 , h 2 , h 3 , h 4 in such a way that the circuit of C incident on v is e, . . . , h 1 , v, h 2 , . . . , h 3 , v, h 4 , . . ., where e ∈ E. Then the fact that P (v) = ψ C (v) indicates that the single transitions at v included in P are {h 1 , h 3 } and {h 2 , h 4 }. It follows that the passages h 1 , v, h 2 ; h 1 , v, h 4 ; h 3 , v, h 2 and h 3 , v, h 4 all have the same "shadow" in T ch(P ). Let us suppose for convenience that the direction of e v in D follows the common shadow of these four passages through v. (If the opposite is true then all the v coordinates mentioned in the next paragraph should be multiplied by −1; but the conclusion of the paragraph after that is unchanged.) The passages of ). That is, the w row of M R, Γ E ( C, P, D) is obtained by subtracting 2 times the v row of M R,Γ E (C, P, D) from the w row of M R,Γ E (C, P, D).
4. Suppose now that C Γ E (w) intersects C Γ E (v) but neither fundamental circuit contains the other. Then , as far as the x coordinates with x / ∈ {v, w} are concerned.
For v and w, though, we have complications similar to the complications at v in part 3. As discussed in part 3, the passages of C Γ E (v) and C Γ E (w) through v have the same shadow in T ch(P ), so the v coordinate of We claim that the w coordinates of z D (C Γ E (w)) − z D (C Γ E (v)) and z D ( C Γ E (w)) are the same, too. To verify this claim we need a detailed analysis of half-edges like the one in part 3. Let h 1 , h 2 , h 3 , h 4 be the half-edges of F incident on w, indexed in such a way that a circuit of C is of the form e, . . . , h 1 , w, h 2 , . . . , h 3 , w, h 4 , . . ., with e ∈ E.
If P (w) = φ C (w) then the single transitions at w included in P are {h 1 , h 2 } and {h 3 , h 4 }. It follows that the passages h 1 , w, h 3 ; h 1 , w, h 4 ; h 2 , w, h 3 and h 2 , w, h 4 all have the same "shadow" in T ch(P ). Let us assume that the direction of e w in D follows this shadow. The passage of 3 , h 4 } are single transitions of P , so the w coordinate of z D (C Γ E (v)) is 0. The passage of C Γ E (w) through w is h 3 , w, h 1 or h 4 , w, h 2 . Either way, the w coordinate of z D ( C Γ E (w)) is −1, the same as the w coordinate of z D (C Γ E (w)) − z D (C Γ E (v)).
If P (w) = χ C (w) then the single transitions at w included in P are {h 1 , h 4 } and {h 2 , h 3 }, so the passages h 1 , w, h 2 ; h 1 , w, h 3 ; h 4 , w, h 2 and h 4 , w, h 3 all have the same "shadow" in T ch(P ). Let us assume that the direction of e w in D follows this shadow. The passage of C Γ E (w) through w is h 3 , w, h 2 ; as {h 2 , h 3 } is a single transition of P , the w coordinate of z D (C Γ E (w)) is 0. The passage of C Γ E (v) through w might be h 1 , w, h 2 or h 3 , w, h 4 ; we refer to these possibilities as subcases (i) and (ii), respectively. The passage of C Γ E (w) through w is h 3 , w, h 1 in subcase (i) and h 4 , w, h 2 in subcase (ii); in either subcase the shadows in T ch(P ) of the passages of C Γ E (v) and C Γ E (w) through w are opposites of each other. Consequently the w coordinates of z D (C Γ E (w)) − z D (C Γ E (v)) and The last possibility to consider is P (w) = ψ C (w). Then the single transitions at w included in P are {h 1 , h 3 } and {h 2 , h 4 }, and the passages h 1 , w, h 2 ; h 1 , w, h 4 ; h 3 , w, h 2 and h 3 , v, h 4 all have the same "shadow" in T ch(P ). We presume that the direction of e w in D follows this shadow. The passage of C Γ E (w) through w is h 3 , w, h 2 , so the w coordinate of z D (C Γ E (w)) is 1. The passage of C Γ E (v) through w is h 1 , w, h 2 or h 3 , w, h 4 , so the w coordinate of z D (C Γ E (v)) is 1. The passage of C Γ E (w) through w is h 3 , w, h 1 or h 4 , w, h 2 , so the w coordinate of z D ( C Γ E (w)) is 0. We see that the w coordinates of z D (C Γ E (w)) − z D (C Γ E (v)) and z D ( C Γ E (w)) are both 0.
Parts 1-4 complete the proof in case P (v) = ψ C (v). As noted early in the argument, In the same way, the case P (v) = φ C (v) includes P (v) = ψ C (v). Consequently the argument for P (v) = φ C (v) is obtained by interchanging C and C in parts 1-4.
It remains to consider the possibility that P (v) = χ C (v). The argument follows the same outline as above, but the details are different in some places. 5 . Then z D (C Γ E (w)) and z D ( C Γ E (w)) are obtained using the same contributions from passages through vertices outside C Γ E (v), and opposite contributions from passages through vertices inside ) as far as passages inside or outside of C Γ E (v) are concerned.
We must still discuss the v coordinate. Again, we index the half-edges incident on v as h 1 , h 2 , h 3 , h 4 in such a way that a circuit of C is e, . . . , h 1 , v, h 2 , . . . , h 3 , v, h 4 , . . ., where e ∈ E. Then the fact that P (v) = χ C (v) indicates that the single transitions at v included in P are {h 1 , h 4 } and {h 3 , h 2 }. It follows that the passages h 1 , v, h 2 ; h 1 , v, h 3 ; h 4 , v, h 2 and h 4 , v, h 3 all have the same "shadow" in T ch(P ); we presume that the direction of e v in D follows the common shadow of these four passages through v. The passages of Just as before, it follows that z D ( C Γ E (w)) = z D (C Γ E (w)) − 2 · z D (C Γ E (v)). 8. If C Γ E (w) intersects C Γ E (v) but neither contains the other then just as before, includes the passage h 1 , v, h 3 through v, so the v coordinate of z D ( C Γ E (w)) is 1 too. On the other hand, if C Γ E (w) includes the passage h 3 , v, h 4 through v then the v coordinates of z D (C Γ E (w)) and z D ( C Γ E (w)) are both −1. We see that the v coordinates of z D ( C Γ E (w)) and z D (C Γ E (w)) − z D (C Γ E (v)) are always the same.
We claim that in addition, the w coordinates of z D ( C Γ E (w)) and z D (C Γ E (w)) − z D (C Γ E (v)) are the same. To verify this claim we can use the same argument as in part 4; the different value of P (v) is irrelevant.
Property II. In the argument above, parts 1 and 5 have z D ( C Γ E (w)) = z D (C Γ E (w)), parts 2 and 6 have z D ( C Γ E (w)) = −z D (C Γ E (w)), parts 3 and 7 have z D ( C Γ E (w)) =

and parts 4 and 8 have
). This is not quite enough as the case P (v) = φ C (v) was not discussed in detail, but instead described by interchanging C and C in parts 1-4. When we interchange C and C in the equalities z D ( C Γ E (w)) = z D (C Γ E (w)) and z D ( C Γ E (w)) = −z D (C Γ E (w)), we obtain the same equalities. When we interchange C and C in z . This is the same as the original equality because part 2 tells us that . Similarly, when we inter- It follows that property II holds when Γ E , Γ E respect the assumption about orientations of fundamental circuits mentioned at the beginning of the argument above. To deal with fundamental circuits that might not respect this assumption, we might have to multiply some rows by −1; but the same rows will require this multiplication for every P . It follows that property II holds when C = C * v, for all choices of Γ E and Γ E . Definition 30. Let C be an Euler system of a 4-regular graph F , and Γ a set of oriented fundamental circuits of C. If v ∈ V (F ) then the "shadow" of C Γ (v) in T ch(C) includes only one edge: C Γ (v) = e v . We denote by D Γ the directed version of T ch(C) in which for each v ∈ V (F ), the direction of e v is chosen so that when we traverse C Γ (v) according to the orientation of C Γ (v), our shadow traverses e v in the D Γ direction.
That is, D Γ is the directed version of T ch(C) with M R,Γ (C, C, D Γ ) = I. Corollary 31. Suppose E is a set of base edges in a 4-regular graph F , C and C are Euler systems of F , and Γ E and Γ E are based sets of oriented fundamental circuits of C and C. Then for every circuit partition P of F and every directed version D of T ch(P ), Proof. Let A be the product of elementary matrices corresponding to the row operations of Theorem 29. Then Theorem 29 tells us that for every circuit partition P and every directed version D of T ch(P ), M R, Γ E ( C, P, D) = A · M R,Γ E (C, P, D). In particular, Corollary 32. Suppose E is a set of base edges in a 4-regular graph F , C and C are Euler systems of F , and Γ E and Γ E are based sets of oriented fundamental circuits of C and C. Then Proof. Taking P = C and D = D Γ E , Corollary 31 tells us that We refer to the formulas of Corollaries 31 and 32 as naturality properties of the M R,Γ E (C, P, D) matrices.
Corollary 33. Let C be an Euler system of a 4-regular graph F , and let Γ E be a based set of oriented fundamental circuits of C. Then det M R,Γ E (C, P, D) ∈ {−1, 0, 1} for every circuit partition P of F and every directed version D of T ch(P ).
Proof. If P is not an Euler system then T ch(P ) has at least one non-loop edge, so the dimension of the cycle space of T ch(P ) is < |V (F )|. According to Theorem 21, this dimension is the same as the rank of M R,Γ E (C, P, D), so det M R,Γ E (C, P, D) = 0.
If P = C is Corollary 33 implies that for based sets of fundamental circuits, Theorem 21 holds over all fields.
Theorem 34. Let C be an Euler system of a 4-regular graph F , and let Γ E be a based set of oriented fundamental circuits of C. Let P be a circuit partition of F and D a directed version of T ch(P ). Then for every field F, the row space of M R,Γ E (C, P, D) over F is equal to the cycle space of T ch(P ) over F.
Proof. Proposition 23 tells us that the row space of M R,Γ E (C, P, D) over F is contained in the cycle space of T ch(P ) over F. To prove that the two spaces are equal, then, it suffices to prove that they have the same dimension.
As Theorem 21 holds over R, the dimension of the cycle space of T ch(P ) over R equals the rank of M R,Γ E (C, P, D) over R. It is well known that the dimension of the cycle space of a graph equals the number of edges excluded from a maximal forest; in particular, this dimension is the same over all fields. Consequently proving the proposition reduces to proving that M R,Γ E (C, P, D) has the same rank over F and R.
The smallest possible value of |P |, the number of circuits in P , is c(F ). If |P | = c(F ) then P is an Euler system of F , and Corollary 33 tells us that det M R,Γ E (C, P, D) = ±1 is nonzero in both F and R. Consequently the rank of M R,Γ E (C, P, D) is n over both F and R.
We proceed using induction on |P | > c(F ). There must be a component of F that contains more than one circuit of P , and this component must contain a vertex v incident on two distinct circuits of P . Let P be a circuit partition that involves the same transitions as P , except that P (v) = P (v). Then P includes the same circuits as P , except that the two circuits of P incident at v are united in one circuit of P . (Two circuit partitions related in this way are pictured in Figure 3.) It follows that T ch(P ) is the graph obtained from T ch(P ) by contracting the edge e v and replacing it with a loop, so the dimension of the cycle space of T ch(P ) is one more than the dimension of the cycle space of T ch(P ).
Let D be the directed version of T ch(P ) in which edge directions are inherited from D. As |P | = |P | − 1, the inductive hypothesis tells us that M R,Γ E (C, P , D ) has the same rank over F and R. The only column of M R,Γ E (C, P , D ) that is not equal to the corresponding column of M R,Γ E (C, P, D) is the v column, so the ranks of M R,Γ E (C, P , D ) and M R,Γ E (C, P, D) cannot differ by more than 1. It follows that the rank of M R,Γ E (C, P, D) over F is at least the dimension of the cycle space of T ch(P ), which is the rank of M R,Γ E (C, P, D) over R.
On the other hand, any relation involving columns of M R,Γ E (C, P, D) over R can be expressed with coefficients from Z, so it reduces to a linear relation involving columns of M R,Γ E (C, P, D) over F. Consequently the rank of M R,Γ E (C, P, D) over R is at least the rank of M R,Γ E (C, P, D) over F. We conclude that M R,Γ E (C, P, D) has the same rank over F and R.
Corollary 35. The nullity of M R,Γ E (C, P, D) over F is equal to |P | − c(F ).
Proof. The dimension of the cycle space of any graph is the difference between the number of vertices and the number of connected components. T ch(P ) has |P | vertices by definition, and it is not hard to see that T ch(P ) has c(F ) connected components. Details are provided in [30, Section 2].

Completing the proof of Theorem 6
Proposition 26 and Theorem 29 imply the following.
Corollary 36. Let E be a fixed set of base edges in a 4-regular graph F , let C and C be oriented Euler systems of F , and let Γ o E and Γ o E be sets of based, consistently oriented fundamental circuits of C and C, respectively. Then IAS Γ o E (C) can be transformed into IAS Γ o E ( C) by a sequence of operations of the following types: multiply a row or column by −1, add an integer multiple of one row to another, or permute columns (in accordance with the φ, χ, ψ labels of elements of T(F ) with respect to C and C).
Proof. At first glance, it might seem that Theorem 29 implies that IAS Γ o E (C) can be transformed into IAS Γ o E ( C) using only row operations, but it is important to remember that if w is a vertex of F then e w may have different directions in D Γ o and D Γ o . This is the case if and only if the half-edges directed toward w + in C Γ o (w) and C Γ o (w) belong to different single transitions in the transition P (v). The columns corresponding to such transitions are the columns we multiply by −1.
As mentioned at the end of Section 5, the only implication of Theorem 6 that has not yet been verified is 1 ⇒ 2. This implication is part of the following.
Corollary 37. Let F be a field, let E be a set that contains one edge from each connected component of a 4-regular graph F , and let Γ o E be a set of based, consistently oriented fundamental circuits of an oriented Euler system C of F . Let M F [IAS Γ o E (C)] denote the matroid represented by the matrix whose entries are the images in F of the entries of We should point out that although the matroid M F [IAS Γ o E (C)] is independent of C, it is not independent of E or F. An example of dependence on E is given in the next section. For dependence on F, note that the fact that IAS Γ o E (C) reduces to IAS(I(C)) modulo 2 implies that M [IAS(G)] = M GF (2) [IAS Γ o E (C)]. As shown in [26], it follows that if G has a connected component with three or more vertices then the matroid is not regular, i.e., it cannot be represented over any field of characteristic = 2. We deduce that M GF (2)

An example
We illustrate the above results with a small example. Figure 4 illustrates two oriented Euler circuits in a 4-regular graph. As in Figure 3, we trace an Euler circuit by walking along the edges of the graph, and maintaining the dashed/plain line status when passing through a vertex. The two illustrated Euler circuits could be represented by the double occurrence words abcdbacd and abcdcabd, respectively.
With E = {ad}, signed versions of these Euler circuits are The resulting matrices are as follows.
As C = C * d, Theorem 29 and Proposition 36 assert that we obtain IAS Γ o {ad} (C) from IAS Γ o {ad} ( C) by using the following operations: permute columns, to reflect the fact that transitions have different φ, χ, ψ formulations with respect to the two Euler circuits; subtract the d row from every other row; multiply the d row by −1; and multiply some columns by −1, to reflect changed edge directions in touch-graphs. We proceed to verify this assertion.
Since φ C (d) = ψ C (d) and χ C (x) = ψ C (x) for all x ∈ {a, b, c} (and the same identities when interchanging C and C), we see that Subtracting the last row from each of the other three yields the next matrix.
To obtain IAS Γ o {ad} (C) multiply the last row by −1, and then multiply every column containing a nonzero entry in the last row by −1.
For later comparison we give now the circuits of cardinality 3 of M R [IAS Γ o {ad} (C)], in a compressed way for readability: The last eight of these listed 3-circuits are transverse circuits; as discussed in [11], they correspond to the eight 3-cycles in F . (Each set of three vertices in F appears on two 3-cycles.) Notice that each of χ C (a), χ C (b), ψ C (c), ψ C (d) appears in precisely three 3-circuits.
If we use one of the cd edges as the base edge instead of ad, then we obtain signed Euler The resulting matrices are given below.
we may use the same sequence of row and column operations given earlier, except that at the end of the sequence only two columns are multiplied by −1. We can also verify that the matroid represented by these matrices over R is not isomorphic to , this matroid has no element that appears in precisely three 3-circuits (in fact, each element appears in an even number of 3-circuits).

Naji's Theorem
In this section we discuss a characterization of circle graphs discovered by Naji [21,22].
Definition 38. Let G be a simple graph. For each pair of distinct vertices v and w of G, let β(v, w) and β(w, v) be distinct variables. Then the Naji equations for G are the following.
2. If v, w, x are three distinct vertices of G such that vw is an edge of G and vx, wx are not edges of G, then β(x, v) + β(x, w) = 0.
3. If v, w, x are three distinct vertices of G such that vw, vx are edges of G and wx is not an edge of G, then β(v, w) + β(v, x) + β(w, x) + β(x, w) = 1.
Theorem 39. ( [21,22,29]) G is a circle graph if and only if the Naji equations of G have a solution over GF (2).
It turns out that the IAS Γ o (C) matrices of Section 5 provide solutions to the Naji equations for the interlacement graph I(C).
Proposition 40. Let C be an Euler system of a 4-regular graph F , and Γ o a set of oriented fundamental circuits of C. Let IAS Γ o (C) = I A B be a corresponding matrix defined as in Section 5. For v = w ∈ V (F ) define β(v, w) ∈ GF (2) as follows.
Proof. The verification is routine. For details see [29,Proposition 4], where the notation v + = v in , v − = v out is used.
The solution described in Proposition 40 cannot be extracted from the GF (2) reduction of IAS Γ o (C), because −1 ≡ 1 and 0 ≡ 2 (mod 2). We come to the rather curious conclusion that even though the Naji equations are defined over GF (2), they are connected to representations of Z 3 (G) over fields of characteristic other than 2.
10 Characterizations of circle graphs and planar matroids in terms of multimatroids In this section we formulate a more detailed form of Theorem 6 in terms of multimatroids.
The power set of a set X is denoted by 2 X . Let Ω be a partition of a set U . A transversal T of Ω is a subset of U such that |T ∩ ω| = 1 for all ω ∈ Ω. The set of transversals of Ω is denoted by T (Ω). A subtransversal of Ω is a subset of a transversal of Ω. The set of subtransversals of Ω is denoted by S(Ω).
In order to efficiently define the notion of a multimatroid, we first recall the notion of a semi-multimatroid.
Definition 41. A semi-multimatroid Z (described by its circuits) is a triple (U, Ω, C), where Ω is a partition of a finite set U and C ⊆ S(Ω) such that for each T ∈ T (Ω), (T, C ∩ 2 T ) is a matroid (described by its circuits).
The elements of C are called the circuits of Z. We say that I ∈ S(Ω) is an independent set of Z is no subset of I is a circuit and we say that B ∈ S(Ω) is a basis of Z if B is an independent set, but no proper superset of B is an independent set. The order of Z is |Ω(Z)|.
For any X ⊆ U , the restriction of Z to X, denoted by Z[X], is the semi-multimatroid (X, Ω , If all elements of Ω(Z) are singletons, then, by slight abuse of notation, we associate Z with the matroid (U (Z), C(Z)). We remark that, unfortunately, the standard way to denote a multimatroid minor (i.e., Z|X) as introduced in [5] clashes with the usual way to denote matroid restriction. To avoid confusion, we denote in this paper the restriction of a matroid M to a subset X of its ground set by M [X] (which is compatible with the notation of multimatroid restriction Z[X]).
Semi-multimatroids Z and Z are called isomorphic if there is a one-to-one correspondence between U (Z) and U (Z ) respecting Ω and C.
For a semi-multimatroid Z and matroid M with ground set U (Z), we say that Z is sheltered by M if Z[T ] = M [T ] for all T ∈ T (Ω(Z)). We say that a matrix A represents (or, is a representation of) a semi-multimatroid Z if A represents a matroid M that shelters Z [10]. We say that a semi-multimatroid Z is representable over some field F if there is a matrix A over F that represents Z. If Z is representable over F, then so is every minor of Z. A semi-multimatroid is called regular if it is representable over all fields. We say that A is a strict representation of Z if A has at most |Ω(Z)| rows. This terminology is consistent with our earlier paper [10], where it is said that M is a strict sheltering matroid if r(M ) ≤ |Ω(Z)|.
Definition 42. A semi-multimatroid Z is called a multimatroid if every minor of Z of order 1 has at most one circuit. A multimatroid is called tight if every minor of Z of order 1 has exactly one circuit.
We remark that we use here the slightly more liberal notion of tightness from [8], while the notion of tightness from [4] additionally requires that no element of Ω(Z) is a singleton.
By definition, both multimatroids and tight multimatroids are closed under taking minors. Also, if Z is a multimatroid and X ⊆ U (Z), then so is Z[X]. However, tight multimatroids are in general not closed under taking restrictions.
If Z is a multimatroid where no element of Ω(Z) is a singleton, then all bases of Z are transversals [4,Proposition 5.5]. If all elements of Ω(Z) for some multimatroid Z are of cardinality q, then Z is called a q-matroid.
It is shown in [10] that for every looped simple graph G, Z 3 (G) is a tight 3-matroid representable over GF (2) and, conversely, every tight 3-matroid representable over GF (2) is isomorphic to Z 3 (G) for some looped simple graph.

Main result
We recall the following known result.
Lemma 43 ( [27]). Let G be a looped simple graph. If H is a vertex-minor of G, then Proof. Let H be a vertex-minor of G. Then there is a graph G locally equivalent to G with G − X = H for some X ⊆ V (G ). By [27,Theorem 2], Z 3 (G ) is isomorphic to Z 3 (G). By [27,Proposition 35], Z 3 (H) is equal to a minor of Z 3 (G ).
Let Z be a minor of Z 3 (G). Then by [27, Proposition 35 and Corollary 38], Z is isomorphic to Z 3 (H) for some vertex-minor H of G.
We say that a strict representation A of Z over R is transversely unimodular if for each transversal T of Z, the square matrix obtained from A has determinant 0, −1, or 1.
We are now ready to prove a detailed form of Theorem 6 in terms of multimatroids.
Theorem 44. Let Z be a tight 3-matroid. Then the following conditions are equivalent.
1. Z has a transversely unimodular strict representation with only integer entries.
3. Z is representable over GF (2) and over some field of characteristic different from 2.
has a transversely unimodular representation (I A) over R, where A is skew-symmetric. Since Z 2 (W 5 ), Z 2 (W 7 ), and Z 2 (BW 3 ) are tight, the restriction to 2-matroids Z 3 (G) − T that are tight is justified. The equivalence of Conditions 4 and 6 in Theorem 44 shows that the skew-symmetry restriction of A can be replaced by the more liberal condition that A has only integer entries (note that the skew-symmetry condition of A along with transversal unimodularity of (I A) implies that A has only integer entries).

A characterization of planarity
This subsection shows that the notion of regularity of tight 3-matroids generalizes the notion of planarity of matroids. It turns out that for every matroid M , Z 2 (M ) is, in fact, a tight multimatroid [5]. We remark that Z 2 (BW 3 ) = Z 2 (F ), where F is the Fano matroid, see, e.g., [16].  [9]. In fact, this holds even when M is quaternary [8]. The uniqueness of Z 3 (M ) follows from the following result.
The next lemma concerns fundamental graphs of binary matroids. More specifically, the bipartite graph G of the next lemma is called a fundamental graph of M .
Lemma 48. Let M be a binary matroid. Then Z 2 (M ) is isomorphic to Z 2 (G) for some bipartite graph G. Moreover, if Z 2 (M ) is isomorphic to Z 2 (G) for some graph G, then G is bipartite.
Proof. If (I A) is a representation of M , then (A T I) is a representation of M * . Hence Z 2 (M ) is sheltered by I A 0 0 0 0 A T I By rearranging columns within skew classes, we obtain the matrix which represent Z 2 (G) for some graph G with 0 A A T 0 as its adjacency matrix. We observe that G is bipartite. Finally, assume Z 2 (M ) is isomorphic to Z 2 (G) for some graph G. We have that Z 2 (G) is represented by φ G (V (G)) χ G (V (G)) I A(G) .
Let f be an isomorphism from Z 2 (M ) to Z 2 (G). Let We now recall that planar matroids correspond to fundamental graphs that are circle graphs.
Proposition 49 (Proposition 6 of [14]). If G is a bipartite circle graph, then Z 2 (G) is isomorphic to Z 2 (M ) for some planar matroid M .
If M is a planar matroid, then Z 2 (M ) is isomorphic to Z 2 (G) for some bipartite circle graph G.
We call a looped simple graph G a looped circle graph if G is obtained from a circle graph by possibly adding some loops.
Proposition 50 (Corollary 43 in [10]). Let G be a looped simple graph. For every transversal T of Z 3 (G), Z 3 (G) − T is isomorphic to Z 2 (G ) for some looped simple graph G locally equivalent to G.
Proof. Corollary 43 in [10] shows that Z 3 (G) − T is isomorphic to Z 2 (G ) for some looped simple graph G . The proof of Corollary 43 in [10] shows that G is actually locally equivalent to G.
The main result of this subsection is the following. Proof. If M is planar, then by Proposition 49 we have that Z 2 (M ) is isomorphic to Z 2 (G) for some circle graph G. By Proposition 47, Z 3 (M ) is isomorphic to Z 3 (G). By Theorem 44, Z 3 (M ) is regular.
Conversely, let Z 3 (M ) be regular. By Theorem 44, Z 3 (M ) is isomorphic to Z 3 (G) for some circle graph G. Thus Z 2 (M ) is isomorphic to Z 3 (G) − T for some transversal T of Z 3 (G). By Proposition 50, Z 2 (M ) is isomorphic to Z 2 (G ) for some looped simple graph G locally equivalent to G. Since G is locally equivalent to circle graph G, G is a looped circle graph (we use here that circle graphs are closed under simple local complement by, e.g., Theorem 15). By Lemma 48, G is bipartite. Thus G does not have any loops. Hence G is a bipartite circle graph. By Proposition 49, Z 2 (G ) is isomorphic to Z 2 (M ) for some planar matroid M . Since Z 2 (M ) and Z 2 (M ) are isomorphic, M is planar too.