The cohesiveness of a point of a graph

The connectivity contribution or cohesiveness of a point v of graph G is defined as the difference k ( G ) ‐ k ( G ‐ v ) where kappa is the usual connectivity symbol. It is shown that if a point v of G has negative cohesiveness, then the set of points adjacent to v is the unique minimum size disconnecting set of G . This theorem has several corollaries including the result that if v has negative cohesiveness in G , then it does not in Γ. Finally we define a cohesiveness triple ( n_, n 0 , n + ) of a graph by taking these, respectively, as the number of negative, zero, and positive cohesiveness points of G . The necessary and sufficient conditions for an arbitrary triple to be the cohesiveness triple of a graph are derived.