Document Type: Research Paper
Department of Mathematics, Faculty of Science, Iran University of Science and Technology, Tehran, Iran.
Let $ G = (V,E) $ be a graph. We say that $ S \subseteq V $ is a defensive alliance if for every $ u \in S $, the number of neighbors $ u $ has in $ S $ plus one (counting $ u $) is at least as large as the number of neighbors it has outside $ S $. Then, for every vertex $ u $ in a defensive alliance $ S $, any attack on a single vertex by the neighbors of $ u $ in $ V-S $ can be thwarted by the neighbors of $ u $ in $ S $ and $ u $ itself. In this paper, we study alliances that are containing a given vertex $ u $ and study their mathematical properties.