WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 18, 2019
(This paper belongs to the Special Issue "Pure Mathematics") On Dual of the Split-off Matroids
Authors: , ,
Abstract: Azadi et al. [Generalization of Splitting off Operation to Binary Matroids, Electronic Notes in Discrete Math, 15 (2003), 186–188] have generalized the splitting off (or in short split-off) operation on graphs to binary matroids. The dual of a split-off matroid is not always equal to the split-off of dual of the original matroid. In this paper, for a given matroid M and two elements x and y from E(M), we first characterize the cobases of the split-off matroid $$M_{x,y}$$ in terms of the cobases of the matroid M. Then, by using the set of cobases of $$M_{x,y}$$ and the set of bases (Azadi characterized this set) of $$(M^*)_{x,y}$$ we characterize those binary matroids for which $$(M_{x,y})^*= (M^*)_{x,y}$$ . Indeed, for a binary matroid M on a set E with $$x, y ∈ E$$, we prove that $$(M_{x,y})^*= (M^*)_{x,y}$$ if and only if $$M = N ⊕ N′$$ where $$N$$ is an arbitrary binary matroid and $$N′$$ is $$U_{0,2}$$ or $$ U_{2,2}$$ such that $$x, y ∈ E(N′)$$.
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Pages: 423-427
WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 18, 2019, Art. #51