Forbidden-minors for graphic and cographic es-splitting matroids

Forbidden-minors for es-splitting binary gammoids

“…(i) is clear from Theorem 1.6. and (ii) follows from Definition 1.5. The proof of (iii) is similar to the proof of Lemma 2.6 [4]. P In the following lemma, we characterize the minimal gammoids with respect to M (K 4 ).…”

“…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). In the similar line Borse [2] gave a forbidden-minor characterization of the class of binary gammoids M such that, for every pair of elements x, y of M , splitting matroid M x,y is a binary gammoid.…”

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

“…In this section, we provide necessary Lemmas which are used in the proof of Theorem 1.6. Dalvi, Borse and Shikare [3] proved the following useful Lemma.…”

“…Let G be a connected graph corresponding to M.Then G has 6 vertices, 10 edges, and has no vertex of degree 2. Hence, by Lemma 2.7, G has minimum degree at least 3 since no two elements are in series.Thus the degree sequence of G is (5,3,3,3,3,3,3) or (4,4,3,3,3,3). By Harary [[7], p 223], each simple connected graph with these degree sequences is isomorphic to one of the graphs of Figure 5 below.…”

“…Dalvi, Borse and Shikare [3,4] characterized graphic (cographic) matroids whose element splitting matroids are graphic (cographic) when |X| = 2. In fact, they proved the following result.…”