On dual of the generalized splitting matroids

Document Type : Research Paper

Authors

Department of Mathematics, Urmia University, Urmia, Iran

10.29252/asta.5.1.81

Abstract

Given a binary matroid $M$ and a subset $T\subseteq E(M)$, Luis A. Goddyn posed a problem that the dual of the splitting of $M$, i.e., ($(M_{T})^{*}$) is not always equal to the splitting of the dual of $M$, ($(M^{*})_{T}$). This persuade us to ask if we can characterize those binary matroids for which $(M_{T})^{*}=(M^{*})_{T}$. Santosh B. Dhotre answered this question for a two-element subset $T$. In this paper, we generalize his result for any subset $T\subseteq E(M)$ and exhibit a criterion for a binary matroid $M$ and subsets $T$ for which $(M_{T})^{*}$ and $(M^{*})_{T}$ are the equal. We also show that there is no subset $T\subseteq E(M)$ for which, the dual of element splitting of $M$, i.e., ($(M^{'}_{T})^{*}$) equals to the element splitting of the dual of $M$, (($M^{*})^{'}_{T}$).

Keywords


[1] S. B. Dhotre, A note on the dual of the splitting matroid, Lobachevskii J. Math., 33, (2012), 229-231.
[2] H. Fleischner, Eulerian Graphs and Related Topics, North Holland, Amsterdam, (1990).
[3] J. G. Oxley, Matroid Theory, Oxford university press, New York, (2011).
[4] T. T. Raghunathan, M. M. Shikare and B. N. Waphare, Splitting in a binary matroid, Discrete math., 184, (1998), 267-271.
[5] M. M. Shikare, Gh. Azadi, Determination of the bases of a splitting matroid, European J. combin., 24, (2003), 45-52.
[6] M. M. Shikare, Gh. Azadi, B. N. Waphare, Generalized splitting operation and its application, J. Indian. Math. Soc., 78, (2011), 145-154.
[7] P. J. Slater, A classsi cation of 4-connected graphs , J. Combin. Theory, 17, (1974), 281-298.
[8] D. J. A. Welsh, Eulerian and bipartite matroids, J. Comb. Theory, 6, (1969), 375-377.