Structural results on delta-matroids and connectivities of graph vertex-minors

Abstract

We present two results about structural properties of delta-matroids which come from graphs and other two results about obtaining vertex-minors and pivot-minors while preserving the connectivities. First, we introduce delta-graphic matroids and study their structural properties to characterize the class of delta-matroids that are twisted matroids and graphic delta-matroids. We give a structural characterization of delta-graphic matroids by providing the decomposition theorem. We also prove that every minor-minimal matroid which is not delta-graphic has at most 48 elements. Second, we introduce Γ-graphic delta-matroids defined from graphs whose vertices are labelled by elements of an abelian group Γ. We provide a polynomial-time algorithm to solve the separation problem on these delta-matroids, and as an application, we present polynomial-time algorithms for two graph problems. We further investigate various properties of Γ-graphic delta-matroids. Third, we show that for any two pairs (Q, R) and (S, T ) of disjoint sets of vertices, if a simple graph is large, then there exist two ways to reduce the graph by a vertex-minor operation while preserving the connectivity between Q and R, and the connectivity between S and T. Finally, we prove that if a sequentially 3-rank-connected graph is large, then there exists a sequen- tially 3-rank-connected vertex-minor with one fewer number of vertices.