Prime vertex-minors of a prime graph

Abstract

A graph is prime if it does not admit a partition (A,B) of its vertex set such that min{|A|,|B|}≥2 and the rank of the A×B submatrix of its adjacency matrix is at most 1. A vertex v of a graph is non-essential if at least two of the three kinds of vertex-minor reductions at v result in prime graphs.In 1994, Allys proved that every prime graph with at least four vertices has a non-essential vertex unless it is locally equivalent to a cycle graph. We prove that every prime graph with at least four vertices has at least two non-essential vertices unless it is locally equivalent to a cycle graph. As a corollary, we show that for a prime graph G with at least six vertices and a vertex x, there is a vertex v≠x such that G∖v or G∗v∖v is prime, unless x is adjacent to all other vertices and G is isomorphic to a particular graph on odd number of vertices.Furthermore, we show that a prime graph with at least four vertices has at least three non-essential vertices, unless it is locally equivalent to a graph consisting of at least two internally-disjoint paths between two fixed distinct vertices having no common neighbors. We also prove analogous results for pivot-minors.