# On the defensive alliances in graph

Document Type: Research Paper

Authors

Department of Mathematics, Faculty of Science, Iran University of Science and Technology, Tehran, Iran.

Abstract

‎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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