A PARTIAL CHARACTERIZATION OF THE COCIRCUITS OF A SPLITTING MATROID

Transcription

1 DEPARTMENT OF MATHEMATICS TECHNICAL REPORT A PARTIAL CHARACTERIZATION OF THE COCIRCUITS OF A SPLITTING MATROID DR. ALLAN D. MILLS MAY 2004 No TENNESSEE TECHNOLOGICAL UNIVERSITY Cookeville, TN 38505

2 A PARTIAL CHARACTERIZATION OF THE COCIRCUITS OF A SPLITTING MATROID ALLAN D. MILLS Abstract. This paper describes some of the cocircuits of a splitting matroid M, in terms of the cocircuits of the original matroid M. 1. Introduction The matroid notation and terminolog used here will follow Ole [2]. In particular, the ground set and the collections of independent sets, bases, and circuits of a matroid M will be denoted b E(M), I(M), B(M), and C(M), respectivel. The fundamental circuit of an element e with respect to the basis B (see [2, p. 18]) will be denoted b C(e, B). Fleischner [1] introduced the idea of splitting a verte of degree at least three in a connected graph and used the operation to characterize Eulerian graphs. For eample, the graph G, in Figure 1 is obtained from G b splitting awa the edges and from the verte v. Raghunathan, Shikare, and Waphare [3] etended the splitting operation from graphs to binar matroids. One of their results [3, Theorem 2.2] can be used to define the splitting operation in a binar matroid in terms of circuits. Definition 1.1. Let M be a binar matroid and suppose, E(M). The splitting matroid M, is the matroid having collection of circuits C(M, )= C 0 C 1 where C 0 = {C C(M), C or, C}; and C 1 = {C 1 C 2 C 1,C 2 C(M),C 1 C 2 =, C 1, C 2 ; and there is no C C 0 such that C C 1 C 2 }. The net result, due to Shikare and Asadi [4], characterizes the bases of a splitting matroid M, in terms of the bases of the original matroid M. Lemma 1.2. Let M be a binar matroid and suppose, E(M). Then B(M, )={B {α} B B(M), α E B and the unique circuit contained in B α contains either or }. The results in the net section describe some of the cocircuits of M, in terms of the cocircuits of M. Recall that the cocircuits of a matroid M 1991 Mathematics Subject Classification. 05B35. 1

3 2 ALLAN D. MILLS v G G, Figure 1. The graph G, is obtained b splitting verte v of G. are the minimal sets having non-empt intersection with ever basis of M. In addition, the basic fact that if C is a circuit and C is a cocircuit of a matroid M, then C C = 1 will be helpful in the proofs. 2. Cocircuits of a Splitting Matroid It follows from Definition 1.1 that if ever circuit of M contains both and, or neither, then C(M, )=C 0 = C(M) and M, = M. The fact that M, M onl if there is a circuit of M containing eactl one of and is the basis of the net two results. Proposition 2.1. If {, } is a cocircuit of M or if {} and {} are cocircuits of M, then M = M,. Proof. In both cases there is no circuit of M containing eactl one of and. Hence M = M,. Proposition 2.2. If eactl one of {, } is a cocircuit of M, then and are cocircuits of M,. Proof. Suppose is a cocircuit of M and is not. Then is in a circuit of M that does not contain. The circuits of M, either contain both and or contain neither nor. Since is in no circuits of M, it follows from the definition of C(M, ) that is in no circuits of M,. Thus is in no circuits of M, and we conclude that is a cocircuit of M,. The previous two results concerned cases in which the set {, } contained a cocircuit of M. The main result of this paper, Theorem 2.4, concerns the case in which {, } is proper subset of a cocircuit of M. Before stating the main result, we first prove the following technical lemma.

4 3 Lemma 2.3. Suppose C is a cocircuit of M and {, } C. Then there eist bases B 1 and B 2 of M such that B 1 (C {, }) = and B 2 (C {, }) = where {, } B 1 = {} and {, } B 2 = {}. Proof. Suppose C is a cocircuit of M and {, } C. It follows from the minimalit of C that there is a basis B of M so that B (C {, }) =. Now suppose ever basis of M having empt intersection with C {, } contains. Then C contains a cocircuit of M; a contradiction. Similarl, if each basis of M having empt intersection with C {, } contains, then C contains a cocircuit of M; a contradiction. We conclude that the lemma holds. Theorem 2.4. Let M, be a splitting matroid obtained from M so that M M,. Suppose {, } is a proper subset of a cocircuit C of M. Then {, } and C {, } are cocircuits of M,. Proof of Theorem 2.4. Suppose {, } is a proper subset of a cocircuit C of M. We first show that {, } is a cocircuit of M,. Since B(M, )={B α B B(M) and C(α, B) contains eactl one of and }, it is clear that {, } has non-empt intersection with each basis of M,. Lemma 2.3 implies that there is a basis B of M so that B, B and B (C {, }) =. Let z C {, }. If C(z,B), then C(z,B) C =1; a contradiction. Then C(z,B), and since C(z,B), it follows that B z is a basis of M,. Moreover, as B z, the set {} is not a cocircuit of M,. Similarl, {} is not a cocircuit of M,. Since {, } is a minimal set having non-empt intersection with each basis of M,, the set {, } is a cocircuit of M,. We now show that the set C {, } has non-empt intersection with each basis of M,. Let B α be an arbitrar basis of M,. If B (C {, }), then clearl (B α) (C {, }). So we ma assume B is a basis of M so that B (C {, }) =. We complete this part of the proof b analzing two cases. First, suppose, B. Now B I(M, ) and B <r(m, )r(m) + 1. So B is a proper subset of a basis B 1 α 1 of M,. Since B 1 α 1 = B α for some α in B 1 B, we ma assume B is a proper subset of the basis B α of M,. Suppose α E(M) (B C ). Then as B α is a basis of M,, the fundamental circuit C(α, B) inm must contain eactl one of and. This implies C(α, B) C = 1; a contradiction. We conclude that α C {, }. Hence (B α) (C {, }) Now suppose B and B. Since B I(M, ) and B <r(m, )= r(m) + 1, there eists α in E(M) B so that B α B(M, ). If C(, B) does not contain, then C(, B) C = 1; a contradiction. Thus C(, B) contains both and. It follows that B B(M, ). Similarl, if α E(M) (B C ), and C(α, B), then C(α, B) C = 1; a contradiction.

5 4 ALLAN D. MILLS So C(α, B) contains neither nor and it follows that B α B(M, ). We conclude that α C {, }. Hence (B α) (C {, }). Therefore each basis of M, must have non-empt intersection with C {, }. We now show that C {, } is a minimal set having non-empt intersection with all bases of M,. Let B be a basis of M so that B, B, and B (C {, }) =. Let z C {, }. If C(z,B) does not contain, then C(z,B) C = 1; a contradiction. Thus C(z,B). Moreover, C(z,B) and it follows that B z B(M, ). Since for all z C {, }, the set B z is a basis of M,, the set C {, } is minimal having non-empt intersection with each basis of M,. We conclude that C {, } is a cocircuit of M,. a d a d b e b e c f c f M(G) M(G(G, ) Figure 2. The matroids M(G) and M(G, ). Theorem 2.4 establishes that if C is a cocircuit of M containing {, }, then C {, } is a cocircuit of M,. We define a Tpe I set of a matroid M to be a set C {, } where C is a cocircuit of M that properl contains {, }. The net table lists the collections of cocircuits of the matroids M and M, shown in Figure 2.

6 5 Cocircuits of M Tpe I sets Cocircuits of M, {d, f} {b} {d, f} {a, b, c} {a, c} {b} {b,, } {a, c} {a,,,c} {, } {c,, e, f} {a,, e, d} {c,, e, d} {a,, e, d} {a,, e, d} {a,, e, f} {a,, e, f} {a,, e, f} {b,, e, d, c} {c,, e, d} {b,, e, f, c} {c,, e, d} {a, b,, e, f} {c,, e, f} {a, b,, e, d} {c,, e, f} Notice that the cocircuits of M, are {, }, the Tpe I sets of M, the sets D X for each cocircuit D of M containing a Tpe I set X, and the cocircuits of M that do not contain a Tpe I set. The following conjecture proposes that this relationship holds in general. Conjecture 2.5. Suppose the splitting matroid M, is obtained from M and {, } is a proper subset of a cocircuit of M. Then {, } C C {, } for each cocircuit C of M properl containing {, } (M, )= D X for each cocircuit D of M containing a Tpe I set X C of M such that C does not contain a Tpe I set acknowledgement This work was partiall supported b a T.T.U. Facult Research Grant. References [1] Fleischner, H., Eulerian Graphs. In Selected Topics in Graph Theor 2 (eds. L.W. Beineke, R.J. Wilson), pp Academic Press, London, [2] Ole, J. G., Matroid Theor, Oford Universit Press, New York, [3] Raghunathan, T. T., Shikare, M. M., and Waphare, B. N., Splitting in a binar matroid, Discrete Mathematics 184 (1998), [4] Shikare, M. M., and Asadi, Ghodratollah, Determination of the bases of a splitting matroid, European Journal of Combinatorics 24 (2003), Mathematics Department, Tennessee Tech. Universit, Cookeville, TN address: amills@tntech.edu