Local majorities, coalitions and monopolies in graphs: a review
Theoretical Computer Science
Discrete Applied Mathematics - Special issue on international workshop on algorithms, combinatorics, and optimization in interconnection networks (IWACOIN '99)
ICCTA '07 Proceedings of the International Conference on Computing: Theory and Applications
Algorithms and complexity for alliances and weighted alliances of various types
Algorithms and complexity for alliances and weighted alliances of various types
Global defensive alliances in star graphs
Discrete Applied Mathematics
Self-stabilizing distributed algorithms for graph alliances
IPDPS'06 Proceedings of the 20th international conference on Parallel and distributed processing
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Given a graph G, a defensive alliance of G is a set of vertices S@?V(G) satisfying the condition that for each v@?S, at least half of the vertices in the closed neighborhood of v are in S. A defensive alliance S is called global if every vertex in V(G)-S is adjacent to at least one member of the defensive alliance S. The global defensive alliance number of G, denoted @c"a(G), is the minimum size around all the global defensive alliances of G. In this paper, we present an efficient algorithm to determine the global defensive alliance numbers of trees, and further give formulas to decide the global defensive alliance numbers of complete k-ary trees for k=2,3,4. We also establish upper bounds and lower bounds for @c"a(P"mxP"n),@c"a(C"mxP"n) and @c"a(C"mxC"n), and show that the bounds are sharp for certain m,n.