LSU Digital Commons LSU Digital Commons Circuit-difference matroids Circuit-difference matroids

One characterization of binary matroids is that the symmetric diﬀerence of every pair of intersecting circuits is a disjoint union of circuits. This paper considers circuit-diﬀerence matroids, that is, those matroids in which the symmetric diﬀerence of every pair of intersecting circuits is a single circuit. Our main result shows that a connected regular matroid is circuit-diﬀerence if and only if it contains no pair of skew circuits. Using a result of Pfeil, this enables us to explicitly determine all regular circuit-diﬀerence matroids. The class of circuit-diﬀerence matroids is not closed under minors, but it is closed under series minors. We characterize the inﬁnitely many excluded series minors for the class.


Introduction
A matroid M is circuit-difference if C 1 C 2 is a circuit whenever C 1 and C 2 are distinct intersecting circuits of M . Evidently, all such matroids are binary. An example of such a matroid is the tipless binary r-spike, that is, the matroid whose binary representation is [I r | J r − I r ], where J r is the r × r matrix of all ones. Subsets X and Y of E(M ) are skew if r(X ∪ Y ) = r(X) + r(Y ). It is easy to check that no two circuits of the tipless binary r-spike are skew. The following is the main result of the paper. To see that this theorem does not extend to all binary matroids, consider the matroid S 8 for which a binary representation is shown in Figure 1. In this matroid, {1, 4, 7, 8} and {2, 3, 5, 6, 8} are circuits whose symmetric difference is the disjoint union of the circuits {1, 2, 6} and {3, 4, 5, 7}. Thus S 8 is not a circuit-difference matroid. However, since r(S 8 ) = 4 and the only 3-circuits of S 8 contain 6, the matroid S 8 has no two skew circuits. Thus one implication of the last theorem fails for arbitrary connected binary matroids. However, as the next result shows, the other implication does hold in the more general context. The proof of this lemma will be given in the next section.  The terminology used here will follow [1] with one exception. In [1], the term "series extension" for matroids is defined as the addition of an element e to a matroid M to create a matroid M in which {e, f } is a cocircuit where f is an element of M , and M /e = M . It will be expedient here to use the term "series extension" more broadly. We shall call a matroid M a series extension of M if it is obtained from M by a sequence of one-element series-extension moves.
Pfeil [2] defined a connected matroid M to be unbreakable if M/F is connected for every flat F of M . He proved that a matroid is unbreakable if and only if its dual has no two skew circuits, and he determined all unbreakable regular matroids. Combining Pfeil's two results gives the following.
Theorem 3. A non-empty connected regular matroid M has no two skew circuits if and only if M is a series extension of one of the following matroids: U 0,1 , U 1,m for some m 1; M * (K n ) for some n 1; M (K 3,3 ); or R 10 .
By combining this theorem with our main result, we have the following explicit description of all regular circuit-difference matroids.
Corollary 4. A regular matroid M is circuit-difference if and only if every component of M is a series extension of one of the following matroids: U 0,1 , U 1,m for some m 1; M * (K n ) for some n 1; M (K 3,3 ); or R 10 .
One can check explicitly, or deduce from the last corollary, that M (K 4 ) is a circuitdifference matroid, but that M (K 4 )/e is not circuit-difference for each element e. Thus the class of circuit-difference matroids is not minor-closed. We shall show that this class is closed under series minors and will characterize the excluded series minors in Section 4. In the next section, we prove some auxiliary results that will be used in the proofs of the main results. The proof of Theorem 1 will be given in the third section.

Preliminary Results
We begin this section with the proof deferred from the last section.
Proof of Lemma 2. Let C 1 and C 2 be skew circuits of M , and let D be a circuit that meets C 1 and C 2 . Since C 1 and C 2 are skew, |D − (C 1 ∪ C 2 )| > 0. As is easily checked, Thus C 1 D meets C 2 D. As their symmetric difference is the disjoint union of the circuits C 1 and C 2 , we deduce that M is not circuit-difference for either C 1 D or C 2 D is not a circuit, or both are circuits but their symmetric difference is not.
The straightforward proof of the next lemma is omitted.
Lemma 5. Let M be a matroid. If M has a pair of skew circuits, then so does every series extension of M .
The next result makes repeated use of the fact that if a circuit in a matroid meets a 2-cocircuit, then it contains that 2-cocircuit. Assume next that e ∈ D 1 − D 2 . Then f ∈ D 1 − D 2 . Now, D 1 − e is a circuit of M /e. Moreover, D 2 is a circuit of M /e as otherwise M would have a circuit that contains e and is contained in D 2 ∪ e. As such a circuit would avoid f , we have a contradiction. We now know that (D 1 − e) D 2 is a circuit of M /e containing f , so D 1 D 2 is a circuit of M .
Finally, assume that e / ∈ D 1 ∪ D 2 . Then f / ∈ D 1 ∪ D 2 and so D 1 and D 2 are circuits of M /e. Hence so is D 1 D 2 . As this set avoids f , it must also be a circuit of M and the lemma is proved.
In the proof of Theorem 1, we will encounter a matroid with the property that the complement of every circuit is a circuit. We call such matroids circuit-complementary. Such matroids that are binary form an interesting subclass of the class of circuit-difference matroids; they are crucial in Section 4 when considering the excluded series minors for the latter class.
Lemma 7. Let M be a connected binary matroid that is circuit-complementary. Then M is a circuit-difference matroid.
Proof. Let C 1 and C 2 be an intersecting pair of circuits of M . Then C 1 C 2 is a disjoint union of circuits. If there are at least two circuits in this union, then, since this union avoids C 1 ∩C 2 , we violate the property that the complement of every circuit is a circuit.
Again, the proof of the next result is elementary and is omitted.   Proof. Let M be M (G), the cycle matroid of the graph G. By Lemmas 2 and 7, M has no two skew circuits. Let C be a cycle of G. Then E(G) − C is a cycle C of G. Now, C and C must have exactly two common vertices, otherwise G is not 2-connected or M has two skew circuits. It follows that G has two vertices u and v that are joined by four internally disjoint paths where these paths use every edge of G.
The following lemma makes repeated use of the fact that in a loopless 2-connected graph, the set of edges meeting a vertex is a bond. Proof. Let M be M * (G), the bond matroid of the graph G. Then G is 2-connected and simple. Take a vertex v of G and let C 1 be the set of edges meeting v. Then C 1 is a bond in G and so is a circuit of M . Thus E(G) − C is also a bond of G. Hence G has a vertex w that is not adjacent to v. Let C 2 be the set of edges meeting w. Then E(G) = C 1 ∪ C 2 and G is isomorphic to K 2,n for some n 2. Let u be a vertex of G other than v or w. The complement of the set of edges meeting u is a bond of G. Thus n = 2 and G is a 4-cycle. Hence M ∼ = U 1,4 .
Lemma 11. Let M be a connected cosimple regular matroid that is circuit-complementary. Then M is isomorphic to U 1,4 or R 10 .
Proof. If M is graphic or cographic, then, by Lemmas 9 and 10, M ∼ = U 1,4 . Now assume that M is neither graphic nor cographic and is not isomorphic to R 10 . Then, by Seymour's Regular Matroids Decomposition Theorem [3], as M is connected, it can be obtained from graphic matroids, cographic matroids and copies of R 10 by a sequence of 2-sums and 3sums. Moreover, each matroid that is used to build M occurs as a minor of M .
If M is not 3-connected, then M has a 2-separation, (X, Y ). Then M is the 2-sum, with basepoint p say, of matroids M X and M Y with ground sets X ∪p and Y ∪p, respectively. As M is connected, so are M X and M Y . Suppose X is independent in M . Then M X must be a circuit with at least three elements. Thus M is not cosimple, a contradiction. We may now assume that both X and Y contain circuits of M . Hence, by the circuit-complementary property, both X and Y are circuits of M . As r(X) + r(Y ) = r(M ) + 1, we see that Then M * is a rank-3 simple binary connected matroid having X and Y as disjoint cocircuits. It follows that M * is graphic, a contradiction. We conclude that 11.1 holds.
We may now assume that there are matroids M 1 and M 2 each with at least seven elements such that E(M 1 ) ∩ E(M 2 ) is a triangle, T , in both matroids, and M is the 3-sum of M 1 and M 2 across this triangle. Moreover, M 1 and M 2 are both minors of M , and As X 1 spans T , it follows that M 1 has rank |X 1 |, so M * 1 has rank three and has T as a triad. Since M is cosimple, no element of X 1 is in a 2-circuit of M * 1 . As M 1 is binary, it follows that |X 1 | 3 so |E(M 1 )| 6, a contradiction. We conclude by the circuit-complementary property that both X 1 and X 2 must be circuits of M . Then, as r(X 1 ) + r(X 2 ) = r(M ) + 2, we have r(M ) = |X 1 | + |X 2 | − 4 so M * has rank four, has (X 1 , X 2 ) as a 3-separation and has each of X 1 and X 2 as a cocircuit. Since M * is a disjoint union of cocircuits, it is affine. As M * is simple, |E(M * )| 8. But |X i | 4 for each i, so |E(M * )| = 8 and M * ∼ = AG (3,2). This contradicts the fact that M is regular and thereby completes the proof of the lemma. Although the following lemma is well known, we include a proof for completeness.
Lemma 13. Let Y be a set in a connected matroid M such that |Y | 2 and M |Y is connected. Let W be a minimal non-empty subset of Proof. This is certainly true if |W | = 1. Now, suppose that w 1 and w 2 are distinct elements of W that are not in series in M |(Y ∪ W ). Then M |(Y ∪ W ) has a circuit K containing w 1 and not w 2 . As W is independent, K meets Y . But K ∩ W ⊆ W − w 2 . Thus we have a contradiction to the choice of W . We deduce that every two elements of W are in series in M |(Y ∪ W ). Since M |Y is connected, no element of Y is in series with an element of W . Thus W is indeed a series class of M |(Y ∪ W ).

The Proof of Main Theorem
In this section, we prove the main result of the paper.
Proof of Theorem 1. Let M be a regular connected matroid. By Lemma 2, if M has a pair of skew circuits, then M is not circuit-difference. To prove the converse, consider all connected regular matroids with no two skew circuits that are not circuit-difference, and choose M to be such a matroid with the minimum number of elements. Then, by Lemma 6, M is cosimple. Let C 1 and C 2 be a pair of intersecting circuits of M such that C 1 C 2 is not a circuit and |C 1 ∪ C 2 | is a minimum among such pairs. As M |(C 1 ∪ C 2 ) is connected, we must have that E(M ) = C 1 ∪ C 2 by our choice of M . Now, C 1 C 2 is a disjoint union of at least two circuits.
Clearly, D meets both C 1 − C 2 and C 2 − C 1 but contains neither of these sets. The choice of {C 1 , C 2 } implies that C 1 D is a circuit and hence that (C 1 D) C 2 is a circuit. The last set is (C 1 C 2 ) − D, so 13.1 holds.
Let Z = C 1 C 2 . As M has no two skew circuits, M |Z is connected and, by 13.1, it is circuit-complementary. Thus, by Lemma 11, M |Z is a series extension of U 1,4 or of R 10 . Let X be a minimal non-empty subset of C 1 ∩ C 2 such that M has a circuit whose intersection with C 1 ∩ C 2 is X. Then, by Lemma 13, X is a series class of M |(Z ∪ X). Thus every circuit of M |(Z ∪ X) that meets X must contain X.

Every circuit of M |Z is a circuit of (M |(Z ∪ X))/X.
Let D be a circuit of M that is contained in Z. Then D meets both C 1 − C 2 and C 2 − C 1 and, by 13.1, Z − D is a circuit of M that also meets both C 1 − C 2 and C 2 − C 1 . Assume that D is not a circuit of (M |(Z ∪ X))/X. Then M |(Z ∪ X) has a circuit K such that K ⊆ D ∪ X and K ∩ D = D. Thus K meets and so contains X. Hence K − D = X. As |K ∪ D| = |X ∪ D| < |C 1 ∪ C 2 |, it follows that K D is a circuit of M and hence that K D meets C 1 − C 2 and C 2 − C 1 .
As |C 1 ∪K| < |C 1 ∪D| < |C 1 ∪C 2 |, the choice of {C 1 , C 2 } implies that C 1 K is a circuit C of M and that C 1 (Z−D) is a circuit C of M . As C and C both contain the non-empty set (D − K) ∩ C 1 and both avoid the non-empty set (D − K) ∩ C 2 , we see that |C ∪ C | < |C 1 ∪ C 2 | and C C is a circuit of M . This circuit is [ . But the last set is a disjoint union of two circuits, a contradiction. Thus 13.2 holds.
We know that M |Z is connected and circuit-complementary. Moreover, the choice of X implies that M |(Z ∪ X) has a circuit that meets C 1 ∩ C 2 in X. Therefore M |(Z ∪ X) is connected. Moreover, by 13.2, (M |(Z ∪ X))/X is connected. It follows by Lemma 12 that (M |(Z ∪ X))/X is not circuit-complementary. Thus (M |(Z ∪ X))/X has a circuit J such that Z − J is not a circuit of (M |(Z ∪ X))/X. If J is a circuit of M |Z, then, as M |Z is circuit-complementary, Z − J is a circuit of M |Z. Thus, by 13.2, we obtain the contradiction that Z − J is a circuit of (M |(Z ∪ X))/X. We deduce that J is not a circuit of M |Z. Then J ∪ X is a circuit, K, of M |Z for some non-empty subset X of X. By the choice of X, it follows that X = X. Now, Z = D ∪ D for some disjoint circuits D and D . We deduce using 13.2 that K meets both D and D but contains neither. Hence is a circuit of M , that is, (Z − K) ∪ X is a circuit of M . Thus Z − J is a circuit of (M |(Z ∪ X))/X, a contradiction.

Excluded Series Minors of Circuit-Difference Matroids
In this section, we show that the class of circuit-difference matroids is closed under series minors, and we characterize the infinitely many excluded series minors for this class. Let N 5 be the 5-element matroid that is obtained from a triangle by adding single elements in parallel to exactly two of its elements. This is easily seen to be an excluded series minor for the class of circuit-difference matroids. Although the next proposition is not needed for the proof of the main result of this section, it seems to be of independent interest.
Proposition 15. A connected binary matroid M has a pair of skew circuits if and only if M has a series minor isomorphic to N 5 .
Proof. If M has a series minor isomorphic to N 5 , then, by Lemma 5, as N 5 has a pair of skew circuits, so does M . For the converse, let C 1 and C 2 be a pair of skew circuits of M , and let D be a circuit meeting both such that |D − (C 1 ∪ C 2 )| is a minimum. Let M = M |(C 1 ∪ C 2 ∪ D). Next we show the following. 15.1.
Suppose that this fails. Then M has a circuit K that contains x but not y. Assume first that K meets C 2 . Then, by the choice of D, we must have that But, for each z in C 1 , the matroid (M |(C 1 ∪ C 2 ))\z has C 2 as its only circuit. As K D = C 2 , we have a contradiction. We deduce that K avoids C 2 . As y / ∈ K, we must have that K ∩ (D − (C 1 ∪ C 2 )) is non-empty. Then K D is a disjoint union of circuits that does not contain D − (C 1 ∪ C 2 ). One such circuit must meet C 2 ∩ D and C 1 . But this violates the choice of D. Thus 15.1 holds.
By 15.1 and symmetry, we can perform a sequence of series contractions in M , reducing each of the sets C 1 − D, C 1 ∩ D, C 2 ∩ D, and C 2 − D to a single element. The resulting matroid is a series minor of M that has two disjoint 2-circuits such that deleting one element from each leaves a circuit with at least three elements. It follows that M has N 5 as a series minor.
We call a matroid hyperplane-complementary if the complement of every hyperplane is a hyperplane. One such matroid is the binary affine geometry AG(r −1, 2) of rank at least two. The next result determines all simple binary hyperplane-complementary matroids. For all k, every rank-k flat of AG(r − 1, 2) is isomorphic to AG(k − 1, 2).
Lemma 16. A simple rank-r binary matroid M is hyperplane-complementary if and only if r 2 and M ∼ = AG(r − 1, 2)\X for some set X such that AG(r − 1, 2)|X does not contain a copy of AG(r − 3, 2).
Proof. Suppose that M is hyperplane-complementary. Then r 2. Moreover, E(M ) is a disjoint union of cocircuits, so every circuit of M has even cardinality. Hence we can view M as AG(r − 1, 2)\X for some set X. Let E = E(AG(r − 1, 2)). Then E(M ) = E − X. Assume that AG(r − 1, 2)|X contains a copy, Z, of AG(r − 3, 2). For y ∈ E − X, consider the closure, cl A (Z ∪ y), of Z ∪ y in AG(r − 1, 2). This closure is a rank-(r − 1) flat of AG(r − 1, 2) and is thus isomorphic to AG(r − 2, 2). Let Y = cl A (Z ∪ y) ∩ (E − X) and W = (E − X) − Y . Then Y is contained in some copy of AG(r − 3, 2), and W is contained in some copy of AG(r − 2, 2). Thus r(Y ) r − 2 and r(W ) r − 1. Hence W is contained in a hyperplane W of M whose complement in E(M ) is not a hyperplane. Thus M is not hyperplane-complementary, a contradiction. Now let M = AG(r − 1, 2)\X where r 2 and AG(r − 1, 2)|X does not contain a copy of AG(r−3, 2). Let H be a hyperplane of AG(r−1, 2). Then AG(r−1, 2)|H = AG(r−2, 2). If r(H − X) r − 2, then H − X is contained in some copy of AG(r − 3, 2) that is contained in H and so, as AG(r − 2, 2) is hyperplane-complementary, X contains a copy of AG(r − 3, 2). This contradiction implies that the hyperplanes of M are all of the sets of the form H − X where H is a hyperplane of AG(r − 1, 2). As AG(r − 1, 2) is hyperplane-complementary, so is M .
Let M be the set of all matroids of rank at least three of the form [AG(r − 1, 2) + e]\X such that AG(r − 1, 2)\X is hyperplane-complementary of rank r. Thus N * 5 is the unique rank-3 member of M while its rank-4 members are the tipped binary 4-spike and a nontip deletion thereof, that is, S 8 . We now show that the duals of the matroids in M are precisely the excluded series minors for the class of circuit-difference matroids. Proof. Let M be an excluded series minor for the class of circuit-difference matroids. By Lemma 6, M is cosimple. Let C 1 and C 2 be intersecting circuits of M such that C 1 C 2 is not a circuit and |C 1 ∪ C 2 | is minimal. 17.1. M * ∈ M.
Evidently, E(M ) = C 1 ∪ C 2 . Then C 1 and C 2 are the only circuits of M containing C 1 − C 2 and C 2 − C 1 , respectively. Now, letting x ∈ C 1 ∩ C 2 , suppose that (C 1 ∩ C 2 ) − x contains an element y. Then, as x and y are not in series, M has a circuit D containing x but not y. As D meets both C 1 and C 2 , we have, by the choice of {C 1 , C 2 }, that C 1 D, C 2 D, and hence (C 1 D) (C 2 D) are circuits of M . This last circuit is C 1 C 2 , so we have a contradiction. Thus C 1 ∩ C 2 = {x}.
To see that M/x is circuit-complementary, suppose that C 1 C 2 has a partition (D 1 , D 2 , . . . , D k ) into circuits of M/x such that k 3. Then each D i meets C 1 −C 2 and C 2 −C 1 . For each i, we see that D i or D i ∪x is a circuit, D , of M . The choice of {C 1 , C 2 } implies that C 1 D , and hence C 1 D C 2 , is a circuit of M . As (C 1 C 2 ) − D i is not a circuit of M/x, we deduce that D = D i for all i. Hence the circuit C 1 D i C 2 of M is the union of k − 1 circuits of M , a contradiction. Thus M/x is circuit-complementary. Therefore, M * \x is hyperplane-complementary. As M is cosimple, M * is simple. Moreover, M * has C 1 −x and C 2 − x as hyperplanes, so M * has the form [AG(r − 1, 2) + e]\X where AG(r − 1, 2)\X is hyperplane-complementary of rank r. Since M * is connected, r(M * ) 2. But if r(M * ) = 2, then M * ∼ = U 2,3 , so M ∼ = U 1,3 and M is circuit-difference, a contradiction. Thus M * ∈ M, so 17.1 holds.
To prove the converse, let M * = [AG(r − 1, 2) + e]\X where AG(r − 1, 2)\X is hyperplane-complementary of rank r and r 3. By Lemma 16, AG(r − 1, 2)|X does not contain a copy of AG(r − 3, 2). Consider AG(r − 1, 2) + e and let H 0 be the hyperplane of P G(r − 1, 2) whose deletion gives AG(r − 1, 2). Take a rank-(r − 2) flat F of P G(r − 1, 2) that is contained in H 0 and avoids e. Apart from H 0 , there are exactly two hyperplanes, H 1 and H 2 , of P G(r − 1, 2) that contain F . Then H 1 − H 0 and H 2 − H 0 are hyperplanes of AG(r−1, 2)+e, and H 1 −(H 0 ∪X) and H 2 −(H 0 ∪X) are hyperplanes of [AG(r−1, 2)+e]\X. The complements of these two hyperplanes are circuits C 1 , C 2 of M that meet in the element e. We now note that C 1 C 2 is not a circuit of M otherwise {e} is a hyperplane of M * , and we obtain the contradiction that r(M * ) 2. Hence M is not circuit-difference. 17.2. If D is a circuit of M \e, then e / ∈ cl(D). Suppose that e ∈ cl(D) for some circuit D of M \e. Then there is a partition {X 1 , X 2 } of D such that X i is a circuit of M/e for both i. As M/e is circuit-complementary having X 1 and X 2 as disjoint circuits, X 1 ∪ X 2 = E(M/e) = E(M ) − e. This is a contradiction as X 1 ∪ X 2 = D E(M ) − e. Hence 17.2 holds. 17.3. M \f is circuit-difference for all f in E(M ).
Suppose some M \f is not circuit-difference. Then it has a pair of intersecting circuits D 1 , D 2 such that D 1 D 2 contains a pair of disjoint circuits K 1 , K 2 . Suppose first that both D 1 and D 2 avoid e. Then so do K 1 and K 2 . Thus, by 17.2, none of D 1 , D 2 , K 1 , or K 2 has e is in its closure. Hence all of D 1 , D 2 , K 1 , and K 2 are circuits of M/e, so M/e is not circuit-difference, a contradiction. Hence at least one of D 1 and D 2 must contain e, so f = e.
Now suppose e ∈ D 1 − D 2 and e ∈ K 1 . Then D 1 − e and D 2 are intersecting circuits of M/e with circuits K 1 − e and K 2 in their symmetric difference. This again contradicts the fact that M/e is circuit-difference. Hence, by symmetry, we must have that e ∈ D 1 ∩ D 2 . Consequently, K 1 , K 2 , D 1 − e, and D 2 − e are circuits of M/e. If D 1 ∩ D 2 = {e}, then D 1 − e and D 2 − e are disjoint. Thus their union is E(M/e). But this union avoids f , a contradiction. Hence (D 1 ∩ D 2 ) − e must be non-empty. But then D 1 − e and D 2 − e are intersecting circuits of M/e so their symmetric difference, which equals D 1 D 2 , is a circuit of M/e. However, this symmetric difference contains K 1 and K 2 , which are circuits of M/e. This contradiction completes the proof of 17.3.
As M * is simple, M is cosimple and hence no series contractions can be performed. Thus, by 17.3, every series minor of M is circuit-difference and the theorem holds.
The next result follows immediately by combining the last theorem with Tutte's excluded-minor characterization of binary matroids [4].
Corollary 18. A matroid M is an excluded series minor for the class of circuit-difference matroids if and only if M ∼ = U n,n+2 for some n 2, or M * can be obtained from AG(r − 1, 2) + e for some r 3 by deleting some set X such that e ∈ X and AG(r − 1, 2)|X does not contain a copy of AG(r − 3, 2).