Global defensive alliances in star graphs

Authors:
Cheng-Ju Hsu;Fu-Hsing Wang;Yue-Li Wang
Affiliations:
Department of Information Management, National Taiwan University of Science and Technology, Taipei, Taiwan, ROC;Department of Information Management, Chinese Culture University, Taipei, Taiwan, ROC;Department of Information Management, National Taiwan University of Science and Technology, Taipei, Taiwan, ROC
Venue:
Discrete Applied Mathematics
Year:
2009

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Abstract

A defensive alliance in a graph G=(V,E) is a set of vertices S@?V satisfying the condition that, for each v@?S, at least one half of its closed neighbors are in S. A defensive alliance S is called a critical defensive alliance if any vertex is removed from S, then the resulting vertex set is not a defensive alliance any more. An alliance S is called global if every vertex in V(G)@?S is adjacent to at least one member of the alliance S. In this paper, we shall propose a way for finding a critical global defensive alliance of star graphs. After counting the number of vertices in the critical global defensive alliance, we can derive an upper bound to the size of the minimum global defensive alliances in star graphs.