University of IsfahanTransactions on Combinatorics2251-86578120190301On the defensive alliances in graph1142322710.22108/toc.2018.50156.1396ENHasan KharaziDepartment of Mathematics, Faculty of Science, Iran University of Science and Technology, Tehran, Iran.Alireza Mosleh TehraniDepartment of Mathematics, Faculty of Science, Iran University of Science and Technology, Tehran, Iran.Journal Article20160317Let $ 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.http://toc.ui.ac.ir/article_23227_49f13d333028b939ed2d096a98282fcf.pdf