On dual of the generalized splitting matroids

Abstract

Given a binary matroid M and a subset T ⊆ E(M ), Luis A. Goddyn posed a problem that the dual of the splitting of M , i.e., ((MT ) * ) 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 (MT ) * = (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 ⊆ E(M ) and exhibit a criterion for a binary matroid M and subsets T for which (MT ) * and (M * )T are the equal. We also show that there is no subset T ⊆ 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 ).