Forbidden-minor characterization for the class of cographic element splitting matroids

Abstract

In this paper, we prove that an element splitting operation by every pair of elements on a cographic matroid yields a cographic matroid if and only if it has no minor isomorphic to M (K 4).

A Forbidden-Minor Characterization for the Class of graphic Matroids which yield the Co-graphic Element-Splitting Matroids

“…It follows from Lemma 2.7 that G can not have (i) more than two edge disjoint triangles; (ii) a cycle of size other than 6 which contains both x and y; and (iii) a triangle and a 2-circuit which are edge disjoint. Then, by case (ii) of Lemma 3.3 of [2], G is isomorphic to G 9 of Figure 4.…”

“…Suppose that G is simple. By Lemma 3.4 of [2], there are 5 non isomorphic simple graphs each with 6 vertices and 12 edges, out of which, two graphs are discarded in case (ii) of Lemma 3.4 of [2]. So, only three graphs are remaining and these graphs are not planar.…”

Forbidden-minors for es-splitting binary gammoids

“…Borse et al [3] gave a forbidden-minor characterization of the class of co-graphic matroids M such that, for every pair of elements x, y of M , splitting matroid M x,y is a co-graphic matroid. Further they [4] characterized graphic (co-graphic) matroids M whose es-splitting matroid M e x,y is also graphic (cographic).…”

Forbidden-minor characterization for the class of graphic element splitting matroids

“…As the matroids F 7 and M (K 5 ) are Eulerian, their dual matroids F * 7 and M * (K 5 ) are bipartite. It is easy to see that a binary matroid M is Eulerian iff the sum of column vectors of A is zero where A is a matrix over GF (2) that represents M. For undefined notation and terminology in graphs and matroids, we refer [6] and [8].…”

“…Then M x,y is said to be obtained from M by splitting away the pair x, y. Various properties concerning the splitting matroid have been studied in [2,7,9,10,11].…”