Cographic Splitting Of Graphic Matroids With Respect To A Set With Three Elements

TL;DR

This paper characterizes when splitting a graphic matroid with respect to a three-element set results in a cographic matroid, extending previous results for two-element sets and exploring properties of binary matroids.

Contribution

It provides a new characterization of graphic matroids that become cographic after splitting with three elements, and offers an alternative proof for the two-element case.

Findings

Characterization of graphic matroids with cographic splits for three-element sets

Extension of previous two-element set results to three-element sets

Alternative proof for the two-element set case

Abstract

In general, the splitting operation on binary matroids does not preserve the graphicness and cographicness properties of binary matroids. In this paper, we obtain a characterization of the class of graphic matroids whose splitting with respect to a set of three elements gives cographic matroids. We also provide an alternate proof to such characterization when the set contains two elements which was provided by Borse et al.