Offensive r-alliances in graphs
Discrete Applied Mathematics
Global defensive k-alliances in graphs
Discrete Applied Mathematics
On the global offensive alliance number of a graph
Discrete Applied Mathematics
On the complement graph and defensive k-alliances
Discrete Applied Mathematics
Computing global offensive alliances in Cartesian product graphs
Discrete Applied Mathematics
Alliance free sets in Cartesian product graphs
Discrete Applied Mathematics
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An offensive k-alliance in a graph is a set S of vertices with the property that every vertex in the boundary of S has at least k more neighbors in S than it has outside of S. An offensive k-alliance S is called global if it forms a dominating set. In this paper we study the problem of partitioning the vertex set of a graph into (global) offensive k-alliances. The (global) offensive k-alliance partition number of a graph @C=(V,E), denoted by (@j"k^g^o(@C)) @j"k^o(@C), is defined to be the maximum number of sets in a partition of V such that each set is an offensive (a global offensive) k-alliance. We show that 2@?@j"k^g^o(@C)@?3-k if @C is a graph without isolated vertices and k@?{2-@D,...,0}. Moreover, we show that if @C is partitionable into global offensive k-alliances for k=1, then @j"k^g^o(@C)=2. As a consequence of the study we show that the chromatic number of @C is at most 3 if @j"0^g^o(@C)=3. In addition, for k@?0, we obtain tight bounds on @j"k^o(@C) and @j"k^g^o(@C) in terms of several parameters of the graph including the order, size, minimum and maximum degree. Finally, we study the particular case of the cartesian product of graphs, showing that @j"k^o(@C"1x@C"2)=@j"k"""1^o(@C"1)@j"k"""2^o(@C"2), for k@?min{k"1-@D"2,k"2-@D"1}, where @D"i denotes the maximum degree of @C"i, and @j"k^g^o(@C"1x@C"2)=max{@j"k"""1^g^o(@C"1),@j"k"""2^g^o(@C"2)}, for k@?min{k"1,k"2}.