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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_, n0, 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.