A chain theorem for sequentially 3-rank-connected graphs with respect to vertex-minors

Abstract

Tutte (1961) proved the chain theorem for simple 3-connected graphs with respect to minors, which states that every simple 3-connected graph G has a simple 3-connected minor with one edge fewer than G, unless G is a wheel graph. Bouchet (1987) proved an analog for prime graphs with respect to vertex-minors. We present a chain theorem for higher connectivity with respect to vertex-minors, showing that every sequentially 3-rank-connected graph G has a sequentially 3-rank-connected vertex-minor with one vertex fewer than G, unless |V(G)|≤12.