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

Abstract

Tutte (1961) proved the chain theorem for $3$-connected graphs with respect to minors, which states that every $3$-connected graph $G$ has a $3$-connected minor with one vertex 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 $|V(G)|\leq 12$.Comment: 21 page