Cohesion and stability in graphs

The cohesion of a vertex v is the minimum number of edges whose deletion makes v a cutvertex of the resulting graph. Such considerations are particularly interesting for alliance and friendship graphs. In this paper, we further examine cohesion, first introduced in [3], with our major goal here being the examination of stability under edge addition. We find necessary and sufficient conditions for a vertex to be stable and prove a theorem which discovers total vertex stability in several graphs.