INTERTWINING CONNECTIVITIES FOR VERTEX-MINORS AND PIVOT-MINORS

Abstract

We show that for pairs (Q, R) and (S, T) of disjoint subsets of vertices of a graph G, if G is sufficiently large, then there exists a vertex v in V(G) -(Q \cup R\cup S \cupT) such that there are two ways to reduce G by a vertex-minor operation that removes v while preserving the connectivity between Q and R and the connectivity between S and T. Our theorem implies an analogous theorem of Chen and Whittle (SIAM J. Discrete Math., 28 (2014), pp. 1402--1404) for matroids restricted to binary matroids.