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 ).