On dual of the generalized splitting matroids

Document Type : Research Paper


Department of Mathematics, Urmia University, Urmia, Iran


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 $M$, (($M^{*})^{'}_{T}$).


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