Vertex cut of a graph and connectivity of its neighbourhood complex

Abstract

We show that if a graph G satisfies certain conditions, then the connectivity of neighbourhood complex N(G) is strictly less than the vertex connectivity of G. As an application, we give a relation between the connectivity of the neighbourhood complex and the vertex connectivity for stiff chordal graphs, and for weakly triangulated graphs satisfying certain properties. Next, we consider graphs with a vertex v such that for any k-subset S of neighbours of v, there exists a vertex vS≠v such that S is subset of neighbours of vS. We prove that for any graph G with a vertex v as above, N(G−{v}) is (k−1)-connected implies that N(G) is (k−1)-connected. We use this to show that:(i) neighbourhood complexes of queen and king graphs are simply connected and (ii) if G is a non-complete stiff chordal graph, then vertex connectivity of G is n+1 if and only if Conn(N(G))=n.