Graphic and cographic Γ-extensions of binary matroids

Abstract

Slater introduced the point-addition operation on graphs to characterize 4-connected graphs. The Γ-extension operation on binary matroids is a generalization of the point-addition operation. In general, under the Γ-extension operation the properties like graphicness and cographicness of matroids are not preserved. In this paper, we obtain forbidden minor characterizations for binary matroids whose Γ-extension matroids are graphic (respectively, cographic).

On graphic elementary lifts of graphic matroids

“…They proved that there are four forbidden minors for this class. However, the authors in [2] observed that one of these four minors is redundant and they restated the result of Shikare and Waphare [16] as follows.…”

“…Theorem 1.2. [2] Let M be a graphic matroid. Then M T is graphic for any T ⊆ E(M ) with |T | = 2 if and only if M has no minor isomorphic to any of the circuit matroids M (G 1 ), M (G 2 ) and M (G 3 ), where G 1 , G 2 and G 3 are the graphs as shown in Figure 1.…”

On k-Connected Γ-Extensions of Binary Matroids

“…Some results on preserving graphicness of M under the Γ-extension operation are obtained in [2]. Borse and Mundhe [6] characterized the binary matroids M for which M X is graphic for any independent set X of M.…”