University of Isfahan Transactions on Combinatorics 2251-8657 8 1 2019 03 01 On the defensive alliances in graph 1 14 23227 10.22108/toc.2018.50156.1396 EN Hasan Kharazi Department of Mathematics, Faculty of Science, Iran University of Science and Technology, Tehran, Iran. Alireza Mosleh Tehrani Department of Mathematics, Faculty of Science, Iran University of Science and Technology, Tehran, Iran. Journal Article 2016 03 17 ‎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. https://toc.ui.ac.ir/article_23227_49f13d333028b939ed2d096a98282fcf.pdf