A note on the dual of the splitting matroid

Abstract

Raghunathan et al. [Splitting in a binary matroid, Discrete Math. 184 (1998), 267–271] have generalized the splitting operation on graphs to binary matroids. Given a binary matroid M and a pair x, y of its elements the dual of the splitting matroid (M* x,y ) need not be same as the splitting of dual matroid (M* x,y ). Luis A. Goddyn (1997, personel communication) posed a problem that dual of a splitting matroid is not always equal to the splitting of dual of the original matroid so, can we characterize those binary matroids, in which both are the same? In this paper, we characterize those binary matroids M for which the two matroids will be same for a given pair of its elements.