On dual of the generalized splitting matroids

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:

Binary matroid , dual of a matroid , n-connected matroid , splitting operation , cocircuit