Isotropic matroids I: Multimatroids and neighborhoods

Several properties of the isotropic matroid of a looped simple graph are presented. Results include a characterization of the multimatroids that are associated with isotropic matroids and several ways in which the isotropic matroid of G incorporates information about graphs locally equivalent to G. Specific results of the latter type include a characterization of graphs that are locally equivalent to bipartite graphs, a direct proof that two forests are isomorphic if and only if their isotropic matroids are isomorphic, and a way to express local equivalence indirectly, using only edge pivots.


Introduction
Let G be a looped simple graph, i.e., a graph in which no two edges are incident on precisely the same set of vertices. Although we allow loops, we reserve the terms adjacent and neighbors for pairs of distinct vertices. The adjacency matrix A(G) is a GF (2)-matrix with nonzero entries on the diagonal for looped vertices, and nonzero entries off the diagonal for adjacent vertices.
In recent work [23], the second author introduced the binary matroid repre- Several properties of isotropic matroids were discussed in [23], including their connection with binary delta-matroids [3] and isotropic systems [2,4], two combinatorial structures studied by Bouchet in the late 1980s as part of the theory of local complementation. In this paper we do not discuss isotropic systems and delta-matroids in detail; instead we consider more general combinatorial structures called multimatroids [7,8,9,10], which were studied by Bouchet in the late 1990s. We recall this notion in Subsection 2.1. Delta-matroids and isotropic systems are both equivalent to particular types of multimatroids. In [13] it is shown that the class of binary delta-matroids can be associated with a subclass of the tight 3-matroids, which in turn constitute a subclass of the class of multimatroids. This has been generalized from binary delta-matroids to quaternary delta-matroids in [14].
The two sets of ideas just summarized are connected through the notion of sheltering matroids, which were mentioned in passing by Bouchet [9]. For a partition Ω of a set U we say that p ⊆ U is a skew pair if |p| = 2 and p ⊆ ω for some ω ∈ Ω. A subset of U that does not contain any skew pair is a subtransversal of Ω, and the set of subtransversals is denoted S(Ω). Notice that if I ∈ S(Ω) is an independent subtransversal of cardinality |Ω|−1, then there is at most one element x / ∈ I such that I ∪ {x} is a dependent subtransversal of Ω. If there is always such an x, and every ω ∈ Ω has |ω| > 1, then we say that Q is tight. Also, we say that a sheltering matroid is a q-sheltering matroid if for all ω ∈ Ω we have |ω| = q. It follows from [23, Proposition 41] that M [IAS(G)] is a tight 3-sheltering matroid, with Ω the partition of W (G) into vertex triples.
In Section 2 we define representability of a sheltering matroid over a field F, and we show that the isotropic matroids of graphs are precisely the GF (2)representable 3-sheltering matroids for which every ω ∈ Ω is a cycle, cf. Theorem 27. The fact that isotropic matroids give rise to tight sheltering matroids is closely related to the fact that isotropic systems give rise to tight multimatroids [9].
In the remainder of the paper we present several results that indicate ways in which the graphical structures of G and locally equivalent graphs are directly related to the matroidal structure of M [IAS(G)]. It follows from the results of Section 2 that these relationships could be equivalently expressed using delta-matroids or isotropic systems, but the explicit definition of the matrix IAS(G) makes it particularly convenient to discuss isotropic matroids.
If v ∈ V (G) then there are three graphs obtained from G in the following ways.

Definition 2 The graph obtained from G by complementing the loop status of
Definition 3 The graph obtained from G by complementing the adjacency status of every pair of neighbors of v is denoted G v s .

Definition 4
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 denoted G v ns .
We say G v ℓ , G v s and G v ns are obtained from G through loop complementation, simple local complementation and non-simple local complementation with respect to v, respectively. A graph that can be obtained (up to isomorphism) from G using these operations is locally equivalent to G. The connection between these operations and isotropic matroids is indicated by the following. Proof. We summarize only the easier "if" direction of the proof from [23].
If H = G v ℓ then the assertion is obvious, as the only difference between IAS(G) and IAS(H) is that the χ(v) and ψ(v) columns are transposed. If H = G v ns the situation is a little more complicated, but the reader can verify without much difficulty that if v is not looped, then IAS(H) can be obtained from IAS(G) by interchanging the φ G (v) and ψ G (v) columns, and then adding the v row to every other row corresponding to a neighbor of v. Elementary row operations do not affect the matroid represented by a matrix, of course, so there is an isomorphism β : The "if" direction follows, as every local equivalence can be expressed as a sequence of loop complementations and non-simple local complementations at unlooped vertices.
We say that an isomorphism β of the type mentioned in Proposition 5 is compatible, or that it is induced by a sequence of loop and local complementations used to obtain H from G.
A transversal of W (G) is a maximal subtransversal, i.e., a subset that includes precisely one element of each vertex triple. The collection of transversals of W (G) is denoted T (G), and the collection of subtransversals (i.e., subsets of transversals) is denoted S(G). If T ∈ T (G) then the restriction M [IAS(G)] | T is a transverse matroid of G. (We use "transverse matroid" to avoid confusion with transversal matroids.) A transverse circuit of G is a circuit of a transverse matroid of G; equivalently, a transverse circuit is an element of S(G) that is a circuit of M [IAS(G)]. Using these ideas we may refine Proposition 5 as follows: Proof. We begin with the implication 3 ⇒ 2. First we recall that a binary matroid is uniquely determined by its cycle space (the span of its circuits under symmetric difference) along with its ground set [19]. Note that every vertex triple is a cycle of the matroid M [IAS(G)]. To verify the implication, it suffices to show that the cycle space of M [IAS(G)] is generated by the vertex triples and the cycles of the transverse matroids. Let C be a cycle of M [IAS(G)], and let O be the set of vertex triples ω with |C ∩ ω| > 1. Then C ′ = C ∆(∆ω∈Oω) is a cycle with |C ′ ∩ ω| ≤ 1 for every vertex triple, so C ′ is a cycle of some (in fact, every) transverse matroid which has C ′ as a subset of its ground set. Hence C = C ′ ∆(∆ω∈Oω) is a sum of transverse cycles and vertex triples. The implication 2 ⇒ 1 is verified in [23]; note that this is stronger than the "only if" direction of Proposition 5 as condition 2 is not restricted to compatible isomorphisms. The implication 1 ⇒ 3 follows directly from Proposition 5, and the implication 4 ⇒ 3 is obvious. The implication 3 ⇒ 4 is almost obvious; we need only observe that if transverse matroids correspond under a bijection between W (G) and W (H), then vertex triples must correspond too.
The relationship between the matroidal structure of M [IAS(G)] and the graphical structure of graphs locally equivalent to G is focused on two particular types of transverse structures in isotropic matroids.
Note that the element of the vertex triple of v included in ζ G (v) corresponds to a column of IAS(G) whose nonzero entries occur in rows corresponding to neighbors of v, so ζ G (v) is a transverse circuit of M [IAS(G)]. Also, G is determined up to isomorphism by the submatroid of M [IAS(G)] whose ground set is Definition 8 Let X be a stable set of G, i.e., X ⊆ V (G) and no two elements of X are neighbors in G. Then By the way, stable sets are often called "independent" but we do not use that term here, to avoid confusion with matroid independence. We should also mention that we include sets of cardinality 0 or 1 among the stable sets.
A looped simple graph G may certainly have transverse circuits that are not neighborhood circuits, and transverse matroids that are not neighborhood matroids. However, it turns out that all transverse circuits and transverse matroids correspond to neighborhood circuits and neighborhood matroids in graphs locally equivalent to G: The following results are direct consequences of Theorems 9 and 10.

Corollary 11
Let G be a looped simple graph, and ν a positive integer. Then G has a transverse matroid of nullity ν if and only if some graph locally equivalent to G has a stable set of size ν. Corollary 12 Suppose G is a looped simple graph, and k ∈ N. Then G has a transverse circuit of size k if and only if some graph locally equivalent to G has a vertex of degree k − 1.
These corollaries can be used to provide particularly simple descriptions of some local equivalence classes. For instance, a computer search using the matroid module for Sage [21,22] indicates that the wheel graph W 5 pictured in Figure 1 is distinctive for relatively small nullities of its transverse matroids (the largest nullity is 2), and relatively large sizes of its transverse circuits (the smallest transverse circuits have 4 elements). These observations yield several characterizations of its local equivalence class: Proposition 13 Let G be a looped simple graph with n ≤ 6 vertices. Then any one of the following properties implies the others.
1. G is locally equivalent to the wheel graph W 5 .
2. G is not locally equivalent to any graph with a vertex of degree ≤ 2.
3. G is not locally equivalent to any graph with a stable set of size ≥ n − 3.
4. G has no transverse circuit of size ≤ 3.
The local equivalence class of W 5 is important in Bouchet's famous characterization of circle graphs by obstructions [6]. In a sequel to the present paper [16] we extend Proposition 13 to provide several new characterizations of circle graphs, using the ideas developed here.
Theorem 9 is proven in Section 3. In Section 4 we provide another instance of the fact that isotropic matroids reflect graph-theoretic properties of locally equivalent graphs, by showing that the isotropic matroid of G detects whether or not G is locally equivalent to a bipartite graph. Theorem 10 and some other results about transverse circuits are deduced from Theorem 9 in Section 5.
In Section 6 we discuss minors of isotropic matroids. Section 7 is focused on a special type of minor: a parallel reduction. It turns out that parallel reductions of isotropic matroids correspond precisely to pendant-twin reductions of graphs and in particular, the graphs whose isotropic matroids can be completely resolved using parallel reductions are the same as the graphs that can be completely resolved using pendant-twin reductions. These are the graphs whose connected components are all distance hereditary [1].
As a special case, in Section 8 we prove the following striking result, which underscores the fundamental difference between isotropic matroids of graphs and the more familiar cycle matroids.

Sheltering matroids
In this section we define the notion of sheltering matroid and show its relationship with the notion of multimatroid from the literature.

Multimatroids
We now recall the notion of multimatroid and related notions from [7]. 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. A transversal q-tuple of a q-partition Ω is a sequence τ = (T 1 , . . . , T q ) of q mutually disjoint transversals of Q.
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 15 ([7])
Let Ω be a partition of a finite set 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 = ∅, I ∪ {x} ∈ I or I ∪ {y} ∈ I.
A multimatroid Z is said to be nondegenerate if |ω| > 1 for all ω ∈ Ω. If Ω is a q-partition, then we say that Z is a q-matroid. If Z is a 1-matroid, then we also view Z simply as a matroid. A basis of a multimatroid Z is a set in I maximal with respect to inclusion. It is shown in [7] that the bases of a nondegenerate multimatroid are of cardinality |Ω|. For X ⊆ U , we define Z[X] = (X, Ω ′ , I ′ ) with Ω ′ = {ω ∩ X | ω ∈ Ω, ω ∩ X = ∅} and I ′ = {I ∈ I | I ⊆ X}. We also define Z − X = Z[U − X]. Moreover, Z is called tight if both |ω| > 1 for all ω ∈ Ω and for every S ∈ S(Ω) with |S| = |Ω| − 1, there is an x ∈ ω such that the rank of the matroid Q[S] (recall that we associate a 1-matroid with a matroid) is equal to the rank of the matroid Q[S ∪ {x}], where ω is the unique set in Ω such that S ∩ ω = ∅.

Sheltering matroids
Recall the notion of sheltering matroid, which was mentioned in the introduction. Many matroid notions carry over straightforwardly to sheltering matroids.
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 [7]. Note that for 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 . If Q 1 and Q 2 are isomorphic then Z(Q 1 ) and Z(Q 2 ) are isomorphic too; but the converse is far from true: Let Z be the multimatroid in which every element of S(Ω) is independent. Then Z has several nonisomorphic sheltering matroids, including the uniform matroids U 4,4 , U 3,4 , U 2,4 and the matroid with bases Note that in the example, the independent sets of the fourth sheltering matroid are all independent in the other three sheltering matroids. This illustrates the following notion.

Definition 18
Let Z be a multimatroid over (U, Ω), which has a sheltering matroid M . Then M is a minimal sheltering matroid of Z if Z has no sheltering matroid N = M whose independent sets are all independent in M .
If (M, Ω) is a sheltering matroid and M is of rank > |Ω|, then a new sheltering matroid Q tr = (M tr , Ω) is obtained by truncation: M tr is the matroid whose independent sets are the independent sets of M of cardinality < r(M ). Clearly Z(Q) = Z(Q tr ). We conclude that for a nondegenerate multimatroid Z sheltered by a matroid M , every minimal sheltering matroid of Z is of rank |Ω|. Example 17 indicates that not all sheltering matroids of rank |Ω| are minimal, though.

Representable multimatroids and sheltering matroids
We say that a sheltering matroid Q = (M, Ω) is weakly representable over the field F if the matroid M is representable over F and M is of rank at most |Ω|. The condition that M is of rank at most |Ω| is equivalent to saying that the family of bases of M that are subtransversals is equal to the family of bases of Z(Q). We call Q weakly representable, because we later define a stronger version of representability for 2-matroids and tight 3-matroids. We say that a multimatroid Z is weakly representable over F if it has a sheltering matroid Q which is weakly representable over F. Note that this notion of weak representability for 1-matroids corresponds with the usual notion of representability for matroids.
One might define an even weaker version of representability for sheltering matroids Q = (M, Ω) (and multimatroids Z) by requiring only that the matroid M is representable over F. One might also consider the notion of krepresentability in-between these two versions of representability, by requiring that the rank of M is at most k ≥ |Ω|. Alternatively, one might require only that Z defines F-representable matroids on the transversals of Ω; Bouchet and Duchamp presented a similar definition in [12]. We do not explore the differences among these weaker versions of weak representability in this paper.
We say that a sheltering matroid and multimatroid are weakly binary when they are weakly representable over GF (2). In this subsection we consider mainly 2-sheltering matroids, and in particular weakly binary 2-sheltering matroids.
Let A be a V × V matrix. The principal pivot transform [24] We do not recall the definition of principal pivot transform, but we note that it has the property that if is a standard representation of some matroid M with respect to basis B and B ′ is another basis of M , then is a standard representation of M with respect to basis B ′ . We say that a matrix A is skew-symmetric if A T = −A. Thus, we allow nonzero diagonal entries if A is over a field of characteristic 2. If A is skew-symmetric, then so is A * X.

Lemma 19 ([7]) Let Ω be a 2-partition of U . Let B ⊆ T (Ω). Then B is the set of bases of a 2-matroid over (U, Ω) if and only if for all
The following lemma is essentially from [3] from the context of delta-matroids. Proof. It suffices to show that Z(Q) is a 2-matroid. We invoke Lemma 19. Let B 1 and B 2 be bases of M and p ⊆ B 1 ∆B 2 be a skew pair. By applying principal pivot transform, we have that M is represented by where q is the skew pair containing q 2 .
We denote Q of Lemma 20 by Q(A, τ, 2). A 2-sheltering matroid Q is said to be representable over some field F if Q = Q(A, τ, 2) for some skew-symmetric matrix A over F and τ some transversal 2-tuple.
We say that a 2-matroid Z is representable over F if there is a 2-sheltering matroid Q representable over F such that Z(Q) = Z. We say that Q (Z, resp.) is binary if Q (Z, resp.) is representable over GF (2). If Q = (M, Ω) is representable over F, then Q is certainly weakly representable over F. We show that the converse holds in case Q is weakly binary and tight. First we prove a lemma.
where ω x is the skew class of Ω containing x ∈ U . Let M be the matroid represented by E. Note that I is an independent set of M . However, there is no x ∈ ω b such that I ∪ {x} is an independent set of M . Thus (M, Ω) is not a 2-sheltering matroid -a contradiction.
Proof. Let -a contradiction of the tightness of Q. Thus A is zero-diagonal. It follows from Lemma 21 that A is symmetric. The proof of Proposition 22 is simple enough, but we should mention that the proposition is closely related to Property 5.2 of Bouchet and Duchamp [12]: if an even delta-matroid is weakly binary, then it is binary. The relationship between the results arises from two facts: if a 2-matroid is weakly binary by our definition, then the associated delta-matroid is weakly binary by their definition; and a binary delta-matroid is even iff the associated 2-matroid is tight. Like the property of Bouchet and Duchamp, Proposition 22 does not hold for weakly binary 2-sheltering matroids in general. In fact, their example S 2 gives us the following example of a weakly binary 2-sheltering matroid that necessarily requires that A be asymmetric.  [12] show that Q is not binary.
The interested reader can verify the observation of Bouchet and Duchamp that Q is not binary in three steps, as follows. First, find all the transverse bases of M ; there are seven, including B and (for instance) B ′ = {a 2 , b 2 , c 1 }. Second, for each of the six transverse bases other than B, find the fundamental circuits of the remaining elements. For instance, the fundamental circuits with respect to The representation of M corresponding to a transverse basis is a GF (2)-matrix of the form I A , where the columns of A are the incidence vectors of the fundamental circuits. The third step is to verify that none of these A matrices is symmetric (presuming the same order of a, b, c is used for the rows and columns). For instance, the A matrix corresponding to B ′ is not symmetric because b 2 ∈ C(a 1 , B ′ ) and a 2 / ∈ C(b 1 , B ′ ).

Lemma 24 For every binary
, then T 1 ∆p with p the skew pair containing x is a basis of exactly one of Q(A, τ, 2) and Q(A ′′ , τ, 2) -a contradiction since Z(Q(A, τ, 2)) = Z(Q(A ′′ , τ ′ , 2)). Consequently, A and A ′′ coincide on the diagonal entries, and so A and A ′′ must differ on some off-diagonal entry. Thus there are x, y ∈ T 2 such that , with p and q the skew pairs containing x and y, is a basis of exactly one of Q(A, τ, 2) and Q(A ′′ , τ, 2) -a contradiction. Therefore, A = A ′′ .

Binary tight 3-matroids and isotropic matroids
We say that a tight weakly binary 3-sheltering matroid Q = (M, Ω) is binary if there is a T ∈ T (Ω) such that the 2-sheltering matroid Q − T is binary. We say that a tight 3-matroid Z is binary if there is a tight binary 3-sheltering matroid Q with Z = Z(Q).
The main results of this subsection are Theorem 26 and Theorem 27 which characterize binary tight 3-matroids and isotropic matroids, respectively.
First we need the following result of [13].
Lemma 25 (Theorems 13 and 16 of [13]) Let Ω be a 3-partition of some finite set U and let T ∈ T (Ω). If Z is a binary 2-matroid over (U \ T, Ω ′ ) with We remark that Theorem 16 of [13] is stated there for delta-matroids rather than 2-matroids; but according to Construction 3.5 of [9] this is not a significant difference.
Moreover, if these statements hold, then Q is uniquely determined by this property.
Proof. Assume the second statement holds, and let G be the looped simple graph whose adjacency matrix is A. We recall from the introduction that M [IAS(G)] is a tight 3-sheltering matroid with Ω the partition of W (G) into vertex triples. Thus Q = (M, Ω) is a tight 3-sheltering matroid. Note that Q is binary since Q − T 3 is binary.
Assume now that the first statement holds. Then Z = Z(Q) for some Q = (M, Ω) such that there is a transversal T 3 such that Q − T 3 is a binary 2-sheltering matroid. Hence Q − T 3 is of the form for some disjoint transversals T 1 and T 2 and some V × V -symmetric matrix A over GF (2). By Lemma 25, there is a unique tight 3-matroid Z ′ over (U, Ω) with Z ′ − T 3 = Z(Q − T 3 ) = Z(Q) − T 3 . Hence Z ′ = Z, and we are done.
Notice that Q = (M, Ω) is isomorphic to an isotropic matroid. Thus, if these statements hold, then we repeat the proof of 3 ⇒ 2 of Theorem 6 to show that M is uniquely determined by the cycles that are subtransversals. Hence M and therefore also Q are uniquely determined by Z.
While Theorem 26 shows that Q is unique with this property, this does not exclude the possible existence of some other binary tight 3-sheltering Q ′ = (M ′ , Ω) with Z(Q) = Z(Q ′ ). The next result shows that this cannot happen if each ω ∈ Ω is a cycle of M ′ , even when Q is weakly binary. This result characterizes isotropic matroids.
1. Q is weakly binary and each ω ∈ Ω is a cycle of M .
2. M is isomorphic to some isotropic matroid where Ω is the set of vertex triples.
Proof. Assume the second statement holds. Recall that for isotropic matroids each vertex triple is a cycle. Also, if M is isomorphic to some isotropic matroid, then Q is obviously weakly binary. Conversely, assume the first statement holds. Since Q is weakly binary and Z(Q) is nondegenerate, M is of rank |Ω| and contains a basis T 1 that is a subtransversal. Let be a standard representation of M with respect to T 1 such that τ = (T 1 , T 2 , T 3 ) is a transversal 3-tuple of Ω. Since each ω ∈ Ω is a cycle of M , the columns belonging to each ω ∈ Ω sum to 0 and so we have B = A + I. By swapping elements from T 2 and T 3 , we may assume, without loss of generality, that each diagonal entry of A is zero. By Lemma 21, A is symmetric since Q − T 3 is a 2-sheltering matroid. Hence M is isomorphic to some isotropic matroid.
In other words, if Q = (M, Ω) is a 3-sheltering matroid where M is binary and of rank |Ω|, and each ω ∈ Ω is a cycle of M , then M is isomorphic to some isotropic matroid (where Ω is the set of vertex triples).
Note that if Q is isomorphic to some isotropic matroid, then Q is tight. Hence, by Theorem 27, if Q is weakly binary and each ω ∈ Ω is a cycle of M , then Q is tight.

Stable sets and transverse matroids
We begin this section with a more detailed version of Proposition 5 of the introduction.
Proposition 28 [23] If G is a looped simple graph with a vertex v then there are for all α ∈ {φ, χ, ψ} and x ∈ V (G), except as follows: In the special cases mentioned in Proposition 28, this vertex bijection does not appear explicitly because it is the identity map of V (G). In general, we use β to denote both a compatible isomorphism of isotropic matroids and the associated vertex bijection; there is little danger of confusing the two, because of the difference between their domains.

Proposition 29 Let G be a looped simple graph, and let J be an independent set of a transverse matroid of G. Then there is a locally equivalent graph H such that only φ H elements appear in the image of J under an induced isomorphism M [IAS(G)] → M [IAS(H)].
Proof. According to part 1 of Proposition 28 we lose no generality if we remove all loops in G; this avoids unnecessary proliferation of cases.
As β v s (J) has only m − 1 non-φ G v s elements, the inductive hypothesis applies.
Suppose instead that every non-φ G element of J is a χ G element. We distinguish two cases. 1. Suppose v and w are neighbors with χ G (v), χ G (w) ∈ J. Then the image of J under the isomorphism β v s : Consequently the argument of the preceding paragraph applies to β v s (J). 2. Suppose Consequently the argument of the preceding paragraph applies to β v s (J). We summarize this special case in the following.

The image of T under β is
Proof. Proposition 29 tells us that there is a graph H locally equivalent to G, such that only φ H elements appear in the image of B under an induced isomorphism β : M [IAS(G)] → M [IAS(H)]. According to part 1 of Proposition 28, this property is not affected if we remove all loops from H, so we may just as well assume that H is a simple graph. We claim that the image of T − B under β cannot include any φ H or ψ H element. Note that the definition of IAS(H) implies that no set of φ H columns can sum to 0, as their nonzero entries appear in different rows. Also, no subtransversal consisting of φ H columns and a single ψ H column can sum to 0, because none of the φ H columns has a nonzero entry in the same row as the diagonal entry of the ψ H column. It follows that every subtransversal of W (H) containing some φ H elements and a single ψ H element is independent. Consequently β(T − B) cannot contain any φ H or ψ H element, because such an element would provide an independent set larger than B.
Suppose now that x, y ∈ V (G) − V B are two vertices of G whose images under β are neighbors in H. Then the χ H (β(x)) column of IAS(H) has a nonzero entry in the β(y) row. As y / ∈ V B , it follows that the χ H (β(x)) column of IAS(H) is not an element of the span of the φ H (β(v)) columns with v ∈ V B . This is impossible, since β(B) spans β(M ). Hence there are no such x and y, Theorem 31 also yields a rather complicated description of the nullity of an arbitrary subtransversal:

An induced isomorphism β : M [IAS(G)] → M [IAS(H)] has
Proof. Suppose the three properties hold, and let T ∈ T (H) be the transversal As X is a stable set in H, the transverse matroid M [IAS(H)] | T is represented by a matrix of the form Here A records adjacencies between vertices in X and vertices in N (X), and I 1 , I 2 are identity matrices. The elements of T corresponding to columns of A and I 1 are all contained in β(S), so |X| is the nullity of β(S) in M [IAS(H)]. Suppose conversely that the nullity of S is ν. Let V S = {v ∈ V (G) | S contains an element of the vertex triple of v}, and let J be an independent subset of S with |S| − ν elements. If M is a transverse matroid of G that contains S, then M has a basis B that contains J. The three properties of the statement follow immediately from Theorem 31, with Notice that in general, choosing a different independent set J will yield a different locally equivalent graph H.

Disjoint transversals and bipartite graphs
If G is a looped simple graph we denote by Φ(G), X(G) and Ψ(G) the transversals of W (G) that include all the φ G , χ G and ψ G elements (respectively). In this section we observe that the union of any pair of disjoint transversals of W (G) provides the 2-sheltering matroid associated with a graph that is locally equivalent to G. We begin with a version of the "triangle property" of isotropic matroids.

Proposition 33 Let T 1 and T 2 be disjoint transversals of W (G). Then every independent subtransversal
Proof. Let J be a maximal independent subtransversal contained in T 1 ∪ T 2 . According to Proposition 29, there is a graph H that is locally equivalent to G, Corollary 35 Let G be a looped simple graph. Then any one of the following conditions is equivalent to the others: 1. G is locally equivalent to a bipartite graph.

G has a pair of disjoint transversals with
Proof. If G is a bipartite graph with vertex-classes V 1 and V 2 , let T 1 and T 2 be the transversals of W (G) given by where I 1 and I 2 are identity matrices. As no row has a nonzero entry in a column corresponding to an element of T 1 and also a nonzero entry in a column corresponding to an element of To verify the implication 1 ⇒ 3 note that if G is locally equivalent to a bipartite graph H, then as was just observed, H has a pair of transversals that satisfy condition 3. The inverse images of these transversals under an induced isomorphism are transversals of G that satisfy condition 3.
To verify the implication 3 ⇒ 2, note that if T 1 and T 2 satisfy condition 3 then r( It remains to verify the implication 2 ⇒ 1. Suppose that G has a pair of disjoint transversals with r(T 1 ) + r(T 2 ) = |V (G)|. By Proposition 33, M [IAS(G)] has a transverse basis B ⊆ T 1 ∪ T 2 . Then B 1 = B ∩ T 1 and B 2 = B ∩ T 2 are both independent sets of M [IAS(G)]; as their cardinalities sum to r(T 1 )+r(T 2 ), each B i must be a maximal independent subset of T i . By Proposition 29, there is a graph H that is locally equivalent to G, such that an induced isomor- As β(B 1 ) ⊆ Φ(H), no column of IAS(H) with a nonzero entry in a row corresponding to a vertex outside V 1 is in the span of the columns corresponding to elements of β(B 1 ). As r(β(T 1 )) = |B 1 |, every column corresponding to an element of β(T 1 − B 1 ) is in the span of the columns corresponding to elements of β(B 1 ); consequently no element of β(T 1 − B 1 ) corresponds to a column that includes a nonzero entry in a row corresponding to an element of V 2 , so no two elements of V 2 are neighbors in H. The same argument applies if we reverse the roles of B 1 and B 2 , so H is a bipartite graph.
For instance, the graph of Figure 2 might at first glance seem to resemble the wheel graph W 5 . But in fact, it is quite different. A computer search indicates that the smallest rank of a transversal of W 5 is 4, but the pictured graph has two disjoint transversals of rank 3. We leave finding them as an exercise for the reader. Here's a hint: local complementations at the degree-2 vertices produce a bipartite graph.

Neighborhood circuits and transverse circuits
Theorem 10 of the introduction follows immediately from Corollary 32, with ν = 1. Corollary 32 is also useful when ν > 1. For instance, the following three results indicate that we can use transverse circuits to detect certain types of vertex pairs in locally equivalent graphs.
Corollary 36 Suppose G is a looped simple graph, and k 1 , k 2 ∈ N. Then these statements are equivalent.
1. G is locally equivalent to some graph H with nonadjacent vertices of degrees k 1 − 1 and k 2 − 1, which do not share any neighbor.
2. G has a transverse matroid with two disjoint circuits of sizes k 1 and k 2 , whose union contains no other circuit.
Proof. Suppose G satisfies condition 1, and let v and w be vertices of H as described.
Then For the converse, let S be the union of the two circuits mentioned in condition 2. Then S is a subtransversal whose nullity is 2. Corollary 32 tells us that there is a graph H that is locally equivalent to G, such that the images of the two circuits mentioned in condition 2 under an induced isomorphism M [IAS(G)] → M [IAS(H)] are both neighborhood circuits.
Corollary 37 Suppose G is a looped simple graph, and k 1 , k 2 ∈ N. Then these statements are equivalent.
1. G is locally equivalent to some graph H with nonadjacent vertices of degrees k 1 − 1 and k 2 − 1, which share a neighbor.
2. G has a transverse matroid of nullity 2, with distinct, intersecting circuits of sizes k 1 and k 2 .
Proof. Let v and w be vertices of a graph H that is locally equivalent to G, as described in condition 1. Then the inverse image of For the converse, let M be a transverse matroid of G of nullity 2, and suppose γ 1 and γ 2 are distinct, intersecting circuits of M with |γ 1 | = k 1 and |γ 2 | = k 2 . The columns of IAS(G) corresponding to elements of γ 1 sum to 0, and so do the columns corresponding to elements of γ 2 . Consequently the columns of IAS(G) corresponding to elements of γ 1 ∆γ 2 also sum to 0. If M were to have a circuit γ 3 γ 1 ∆γ 2 , then it would also have a circuit γ 4 ⊆ (γ 1 ∆γ 2 ) − γ 3 , because the columns of IAS(G) corresponding to elements of (γ 1 ∆γ 2 ) − γ 3 would sum to 0. Then an independent set J of M would have to exclude an element x of γ 3 and an element y of γ 4 , and at least one more element: if x, y ∈ γ 1 − γ 2 then J would have to exclude an element z of γ 2 , if x, y ∈ γ 2 − γ 1 then J would have to exclude an element z of γ 1 , and if x ∈ γ 1 − γ 2 and y ∈ γ 2 − γ 1 then J would have to exclude some element z of γ 1 ∪ γ 3 − {x}, as the circuit elimination property guarantees that γ 1 ∪ γ 3 − {x} is dependent. As the nullity of M is only 2, we conclude by contradiction that γ 1 ∆γ 2 is a circuit of M .
Let J be a subset of M that excludes an element of γ 1 − γ 2 and also excludes an element of γ 2 −γ 1 . Then J is an independent set of M [IAS(G)], and applying Corollary 32 with this choice of J tells us that there is a graph H that is locally equivalent to G, such that the images of Corollary 38 Let G be a looped simple graph, and let k 1 , k 2 ∈ N. Then these statements are equivalent.
1. G is locally equivalent to a graph with adjacent vertices of degrees k 1 − 1 and k 2 − 1.
Proof. Suppose G is locally equivalent to a graph H with adjacent vertices v 1 and v 2 , of degrees k 1 − 1 and k 2 − 1. Then the neighborhood circuits ζ H (v 1 ) and One independent subtransversal of maximum size contains only φ H elements, and the other includes one of

Matroid minors and vertex-minors
Isotropic matroids of graphs constitute a very limited class of binary matroids. The limitation is clear even if we note only that they are 3n-element matroids, as this implies that when a single element is contracted or deleted from an isotropic matroid, the result cannot be an isotropic matroid.
There is a special minor operation that is appropriate for isotropic matroids, which involves removing entire vertex triples.
Definition 39 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 Notice that if S is specified then it is not necessary to explicitly mention S ′ , as S ′ is determined by S. Consequently we may sometimes simply refer to the isotropic minor obtained by contracting S. By the way, the definition is consistent with Bouchet's definitions of minors of isotropic systems [2] and multimatroids [8].
Definition 40 A vertex-minor of a looped simple graph G is a graph obtained from G through some sequence of local complementations, looped complementations and vertex deletions.
Theorem 41 [23,Section 7.1] The isotropic minors of G are precisely the isotropic matroids of vertex-minors of G.
In particular, if v ∈ V (G) is unlooped and w ∈ N G (v) then the isotropic minor of G obtained by contracting Notice that we only say "is isomorphic to" in the first case, because in that case the matroid isomorphism requires a permutation of the φ, χ, ψ labels in the vertex triple of w. No such label change is needed in the other two cases. We refer to [23] for details.
In contrast, it turns out that all minors of transverse matroids are transverse minors, in an appropriate sense. It is regrettable that the most important vertex-minor-closed family of looped simple graphs, the looped circle graphs, cannot be described in this easy way. (Looped circle graphs constitute a proper subfamily of G cographic .) We discuss this important family in a sequel to the present paper [16].

Parallel reductions and distance hereditary graphs
Recall some elementary definitions of matroid theory. A loop of a matroid is an element that is excluded from every basis. Two non-loop elements x and y are parallel if {x, y} is a circuit; equivalently, no basis includes them both. We also consider all loops to be parallel to each other. Dually, a coloop is an element that is included in every basis, and we consider all coloops to be in series with each other. Two non-coloop elements are in series if no basis excludes them both.
It is a simple matter to recognize loops and parallels in matroids represented by matrices: a column represents a loop if all of its entries are 0, and two columns represent parallels if all of their entries are the same. In general it is not quite so easy to recognize coloops and elements in series, but this will not concern us because isotropic matroids contain no series pairs that are not also parallel: , so at least one of σ, τ must be included in Φ(G); say σ = φ G (v) ∈ Φ(G). Suppose v is not isolated in G, and let x be a neighbor of v. Then χ G (x) and ψ G (x) both have nonzero entries in the v row; choose one of these two that is not equal to τ , and denote it σ ′ . Then Φ(G)∆{σ, σ ′ } is a basis of M [IAS(G)], which contains neither σ nor τ ; this contradicts the assumption that σ and τ are in series. Consequently v must be isolated.
If τ is the other element of the vertex triple of v that is not a loop in M [IAS(G)], then the columns of IAS(G) corresponding to σ and τ are identical, and no other column of IAS(G) has a nonzero entry in the v row of IAS(G). Hence σ and τ are both parallel and in series.
Suppose τ is an element of the vertex triple corresponding to a vertex w = v. Let σ ′ be the non-loop element of the vertex triple of v, other than σ. If τ = φ G (w) then Φ∆{σ, σ ′ } is a basis of M [IAS(G)], which excludes σ and τ . If τ = φ G (w), let τ ′ be the other element of the vertex triple of w with a nonzero entry in the w row. Then Φ(G)∆{σ, σ ′ , τ, τ ′ } is a basis that excludes σ and τ . In either case, we have a contradiction.
In contrast, there are several kinds of parallels in isotropic matroids.
is an example of an automorphism that satisfies part 2(a).
If case 2 or case 3 of Proposition 46 holds then as noted before the statement of this corollary, G is locally equivalent to a graph H in which case 4 of Proposition 46 holds. Let β : M [IAS(G)] → M [IAS(H) be a compatible isomorphism induced by a local equivalence between G and H. We have just seen that there are automorphisms β a and β b of M [IAS(H)], which satisfy parts 2(a) and 2(b) of the statement for H (respectively). It follows that the compositions β −1 β a β and β −1 β b β satisfy the statement for G.
The familiar idea of parallel reduction in matroid theory is simply to delete one of a pair of parallels. It makes little difference which of the two parallels is deleted, because the identity map of the ground set defines an isomorphism between the two resulting matroids. We would like to define an analogous notion of "parallel reduction" for isotropic matroids, but regrettably it cannot be quite so simple. There are two complications here that do not affect ordinary matroidal parallel reduction: • To obtain an isotropic minor of an isotropic matroid we cannot simply delete an element. We must remove a whole vertex triple, by deleting two elements and contracting the third.
• Corollary 47 tells us that choosing which parallel to delete from M [IAS(G)], and which element of that vertex triple to contract, will not affect the resulting isotropic matroid up to isomorphism. However, such an isomorphism need not be defined by the identity map of W (G). For instance, if v is unlooped and w ∈ N (v) then as noted in connection with Theorem 41, when we contract χ G (v) the resulting isotropic minor is isomorphic to , and an isomorphism involves changing the φ, χ, ψ designations of some matroid elements. On the other hand, if we contract φ G (v) then the resulting isotropic minor is identical to M [IAS(G − v)].
Considering these complications, we always prefer to contract a φ element; consequently we always prefer to delete a parallel that is not a φ element. (Note that it is impossible for two parallels to both be φ elements, because no two columns of an identity matrix are the same.) The following definition reflects these preferences.
Definition 48 Let G be a looped simple graph, and suppose σ and τ are distinct, parallel elements of M [IAS(G)] such that σ is not a φ G element. An isotropic parallel reduction of M [IAS(G)] corresponding to the pair σ, τ is an isotropic minor obtained by contracting the φ G element of the vertex triple that contains σ, and deleting both σ and the third element of that vertex triple.
Definition 49 Let G be a looped simple graph. A pendant-twin reduction of G is a graph obtained from G in one of the following ways: If G satisfies Corollary 51 then we refer to the two sequences of reductions as resolutions, the first a pendant-twin resolution of G and the second an isotropic parallel resolution of M [IAS(G)]. A connected graph that admits such resolutions is called distance hereditary [1]. Corollary 51 should be viewed as an extension of the discussion of these graphs by Ellis-Monaghan and Sarmiento [18], who proved that if a distance hereditary graph has a pendant-twin resolution without any adjacent twin reduction, then it is the interlacement graph of a medial graph of a series-parallel graph. Corollary 51 gives us a matroidal characterization of arbitrary distance hereditary graphs: M [IAS(G)] has an isotropic parallel resolution if and only if the connected components of G are all distance hereditary.

Forests
Theorem 14 follows directly from two results that are already known. One is the equivalence between parts 1 and 2 of Theorem 6, and the other is a theorem of Bouchet [5], who verified a conjecture of Mulder by proving that locally equivalent trees are isomorphic. Bouchet's proof of Mulder's conjecture involves Cunningham's theory of split decompositions [17]; we provide an alternative argument that involves isotropic parallel reductions instead.
The first step in this alternative argument is a special case of Proposition 46. Consequently the inductive hypothesis tells us that G − v ∼ = H − β(v). As v and β(v) are both isolated, it follows that G ∼ = H. Moreover, this graph isomorphism is given by a bijection that agrees with the isomorphism G−v ∼ = H −β(v) given by the induction hypothesis, and it matches v to β(v), so it satisfies the strong form of the theorem.
Then β induces an isomorphism between the isotropic minors The inductive hypothesis tells us that there is a bijection between V (G − v) and V (H − β(v)), which defines a graph isomorphism and agrees with the bijection defined by β at every vertex x ∈ V (G − v) where β(φ G (x)) = φ H (β(x)). In particular, the isomorphism matches w to β(w). As v and β(v) are pendant on w and β(w) respectively, it follows that we can extend that isomorphism to an isomorphism G ∼ = H, which matches v to β(v). Clearly this isomorphism also satisfies the strong form of the theorem.