On the defensive alliances in graph

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.


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