Non-separating cocircuits in matroids

Manoel Lemos 1,3

Departamento de MatematicaUniversidade Federal de Pernambuco

Recife-PEBrazil

T. R. B. Melo 2,4

Departamento de MatematicaUniversidade Federal de Pernambuco

Recife-PEBrazil

Abstract

For a 3-connected binary matroid M , let dimA(M) be the dimension of the subspaceof the cocycle space spanned by the non-separating cocircuits of M avoiding A,where A ⊆ E(M). When A = ∅, Bixby and Cunningham, in 1979, showed thatdimA(M) = r(M). In 2004, when |A| = 1, Lemos proved that dimA(M) = r(M)−1.Recently Lemos and Melo proved that if M is a 3-connected regular matroid andA is a 2-subset of E(M), then 2dimA(M) ≥ r(M)− 3. In this paper, we show thatif M is a 3-connected binary matroid and A is any subset of E(M) such that M |Ahas no coloop, then dimA(M) ≥ r(M) − (2|A| − |A| − 1). We also show that whenA is a triangle of a 3-connected binary matroid M , then dimA(M) = r(M) − 2.

Keywords: binary matroid, cocircuit, connected hyperplane, hyperplane, matroid,non-separating cocircuit

Electronic Notes in Discrete Mathematics 19 (2005) 149–153

1571-0653/2005 Published by Elsevier B.V.

www.elsevier.com/locate/endm

doi:10.1016/j.endm.2005.05.021

1 Introduction

We say that a cocircuit C∗ of a matroid M is non-separating when M\C∗ isconnected. Note that a cocircuit of a matroid M is non-separating if and onlyif its complement is a connected hyperplane of M . For a connected graphicmatroid, a non-separating cocircuit corresponds to the star of a vertex whosedeletion from the associated graph keeps it 2-connected.

Non-separating circuits and cocircuits play an important role in the un-derstanding of the structure of graphic matroids. For example, with the aidof these cocircuits, Kelmans [4] gave an elegant demonstration of Whitney’s2-Isomorphism Theorem (see [13]) and Tutte [11] obtained a nice characteri-zation of the 3-connected graphs which are planar. Bixby and Cunningham[1] generalized Tutte’s result for the class of binary matroids by proving Ed-monds’s Conjecture, namely: a 3-connected binary matroid is graphic if andonly if each element belongs to exactly two (or at most two) non-separatingcocircuits. Moreover, Bixby and Cunningham also proved that each elementof a 3-connected binary matroid belongs to at least two non-separating cocir-cuits. Kelmans [3] and, independently, Seymour (see [9]) proved that everysimple and cosimple connected binary matroid has a non-separating cocircuit.It is somewhat striking that every connected binary matroid which is simpleand cosimple has at least four non-separating cocircuits as proved by McNultyand Wu [7]. Moreover, McNulty and Wu’s result is sharp: there is an infinityfamily of matroids that attains the bound. In general, even a 3-connectedmatroid may not have a non-separating cocircuit. This is true, for example,for Ur,n provided 2 < r < n. (For matroid notation and terminology, we followOxley [10].)

The problem of finding non-separating cocircuits of a simple and cosimpleconnected binary matroid was reduced by Lemos and Melo [6] to the prob-lem of finding non-separating cocircuits of some 3-connected binary matroidsavoiding a set of elements. Let R∗

A(M) be the set of non-separating cocircuitsof a matroid M avoiding a subset A of E(M). (When A = ∅, we use R∗(M)instead of R∗

A(M).) If M is a simple and cosimple binary matroid, then

R∗(M) = R∗A1

(M1) ∪R∗A2

(M2) ∪ · · · ∪ R∗Am

(Mm),(1)

1 Partially supported by CNPq (Grants No. 302195/02-5 and 664107/97-4) andFAPESP/CNPq (Grant No. 2003/09925-5).2 Supported by CNPq (Grant No. 140133/00-3) and FACEPE (Grant No.BFD03981.01/2004).3 Email:manoel@dmat.ufpe.br4 Email:trbm@dmat.ufpe.br

M. Lemos, T.R.B. Melo / Electronic Notes in Discrete Mathematics 19 (2005) 149–153150

where M1,M2, . . . ,Mm are the 3-connected matroids that label the vertices ofthe canonical tree decomposition TM of M and, for i ∈ {1, 2, . . . ,m}, Ai =E(Mi) − E(M). (For a definition of TM see [2] or [5].)

From (1), when M is a simple and cosimple connected binary matroid,

|R∗(M)| =m∑

i=1

|R∗Ai

(Mi)|.(2)

To obtain a lower bound for |R∗(M)|, it is enough to find one for |R∗Ai

(Mi)|,for some i ∈ {1, 2, . . . ,m}. To describe the known bounds for |R∗

Ai(Mi)|, we

need to give some definitions.

For some n > 2, let On be the vector matroid of the matrix [In|Xn] overGF (2), where Xn = (xij) is an n × n matrix such that

xij =

⎧⎨⎩

0, when i = j and 1 ≤ i ≤ n − 1;

1, when i �= j or i = j = n..

Note that On is a 3-connected matroid. If the 2n columns of [In|Xn] are labeledby a1, a2, . . . , an−1, a, b1, b2, . . . , bn−1, b respectively, then R∗

{a,b}(On) = ∅.For some n > 2, label the maximal stable sets of K3,n by V1 and V2 so

that |V1| = 3. Let K ′′3,n be graph obtained from K3,n by adding two pairwise

non-parallel edges joining vertices in V1.

Let M be a 3-connected binary matroid. For a subset A of E(M), we de-note by dimA(M) the dimension of the subspace of the cocycle space spannedby the non-separating cocircuits of M avoiding A. When A = ∅, we usedim(M) instead of dimA(M). Note that dimA(M) is a lower bound for|R∗

A(M)|. All the lower bounds for dimA(M) given by the next result aresharp.

Theorem 1.1 Let M be a 3-connected binary matroid. If A is a subset ofE(M), then

(i) (Bixby and Cunningham [1]) dimA(M) = r(M), when |A| = 0.

(ii) (Lemos [5]) dimA(M) = r(M) − 1, when |A| = 1.

(iii) (Lemos and Melo [6]) dimA(M) ≥ r(M)+1−n2

, when |A| = 2 and M hasno minor isomorphic to On, for some n > 2.

(iv) (Lemos and Melo [6]) dimA(M) ≥ r(M) + 1 − n, when |A| = 2 and Mhas no minor isomorphic to On or to M ∗(K ′′

3,n−1), for an integer n > 2.

Using (2) and Theorem 1.1(ii), one can obtain McNulty and Wu [7] bound

M. Lemos, T.R.B. Melo / Electronic Notes in Discrete Mathematics 19 (2005) 149–153 151

for the number of non-separating cocircuits of a simple and cosimple bi-nary matroid M which is not 3-connected, since there is at least two i ∈{1, 2, . . . ,m} such that |Ai| = 1. (That is, TM has at least two terminal ver-tices.) When M is a 3-connected binary matroid, Bixby and Cunningham [1]proved that M has at least r(M) + 1 non-separating cocircuits. Observe thatthere is a huge gap between the bounds for the number of non-separating co-circuits obtained by McNulty and Wu [7], for a simple and cosimple connectedbinary matroid, and by Bixby and Cunningham [1], for a 3-connected binarymatroid. Lemos [5] proved that M has at least r(M) − 1 non-separating co-circuits, when M is a simple and cosimple connected binary matroid havingjust one 2-separation. He also constructed an infinity family of matroids toshow that the bound given by McNulty and Wu [7] is sharp when the matroidhas exactly two 2-separations.

The next result is a consequence of Theorem 1.1(iii), for n = 4, becauseO4 is not regular.

Corollary 1.2 Let M be a 3-connected regular matroid. If A is a 2-subset ofE(M), then

|R∗A(M)| ≥ dimA(M) ≥

⌈r(M) − 3

2

⌉.

2 Main Results

When M is graphic and M |A has no coloop, it is easy to see that

dimA(M) ≥ r(M) − (|A| − 1).(3)

Moreover, this bound is sharp. In general, we find a lower bound that is closedto (3), namely:

Theorem 2.1 Let M be a 3-connected binary matroid. If A is a subset ofE(M) such that M |A has no coloop, then

|R∗A(M)| ≥ dimA(M) ≥ r(M) − (2|A| − |A| − 1).

These two bounds have the same behavior when r(M) goes to infinity andA is fixed. We think that the bound in Theorem 2.1 is far from being sharp,but when A is a triangle of M , we have the following result:

Theorem 2.2 Let M be a 3-connected binary matroid. If A is a triangle ofM , then

|R∗A(M)| ≥ dimA(M) = r(M) − 2.

M. Lemos, T.R.B. Melo / Electronic Notes in Discrete Mathematics 19 (2005) 149–153152

This paper is based on part of the Ph.D. thesis of Melo [8].

References

[1] R. E. Bixby and W. H. Cunningham, Matroids, graphs, and 3-connectivity, in“Graph Theory and Related Topics” (J. A. Bondy and U. S. R. Murty, Eds.),pp. 91-103, Academic Press, New York, 1979.

[2] W. H. Cunningham, A combinatorial decomposition theory, Ph. D. Thesis,University of Waterloo, 1973.

[3] A. K. Kelmans, The concepts of a vertex in a matroid, the non-separatingcircuits and a new criterion for graph planarity, in “Algebraic Methods in GraphTheory, Vol. 1”, Colloq. Math. Soc. Janos Bolyai (Szeged, Hungary, 1978), Vol.25, pp 345-388, North Holland, Amsterdam, 1981.

[4] A. K. Kelmans, Graph planarity and related topics, in “Graph StructureTheory” (N. Robertson and P. D. Seymour, Eds.), Contemporary Mathematics,Vol. 147, pp. 635-667, 1991.

[5] M. Lemos, Non-separating cocircuits in binary matroids, Linear Algebra Appl.382 (2004), 171-178.

[6] M. Lemos and T. R. B. Melo, Connected hyperplanes in binary matroids,submitted.

[7] J. McNulty and H. Wu, Connected hyperplanes in binary matroids, J. Combin.Theory Ser. B 79 (2000), 87-97.

[8] T. R. B. Melo, Hiperplanos Conexos em Matroides Binarias, Tese de Doutorado,Universidade Federal de Pernambuco, Recife, 2004.

[9] J. G. Oxley, Cocircuit coverings and packings for binary matroids, Math. Proc.Cambridge Phil. Soc. 83 (1978), 347-351.

[10] J. G. Oxley, Matroid Theory, Oxford Univ. Press, New York, 1992.

[11] W. T. Tutte, How to draw a graph, Proc. London Math. Soc. 13 (1963), 734-768.

[12] W. T. Tutte, Connectivity in matroids, Canad. J. Math. 18 (1966), 1301-1324.

[13] H. Whitney, 2-isomorphic graphs, Amer. J. Math. 55, 245-254.

[14] H. Wu, On vertex-triads in 3-connected binary matroids, Combin. Probab.Comput. 7 (1998), 485-497.

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