Strategic balance in graphs

Abstract

For a given graph G, a nonempty subset S contained in V(G) is an alliance iff for each vertex v∈S there are at least as many vertices from the closed neighbourhood of v in S as in V(G)−S. An alliance is global if it is also a dominating set of G. The alliance partition number of G was defined in Hedetniemi et al. (2004) to be the maximum number of sets in a partition of V(G) such that each set is an alliance. Similarly, in Eroh and Gera (2008) the global alliance partition number is defined for global alliances, where the authors studied the problem for (binary) trees.In the paper we introduce a new concept of strategic balance in graphs: for a given graph G, determine whether there is a partition of vertex set V(G) into three subsets N, S and I such that both N and S are global alliances. We give a survey of its general properties, e.g., showing that a graph G has a strategic balance iff its global alliance partition number equals at least 2. We construct a linear time algorithm for solving the problem in trees (thus giving an answer to the open question stated in Eroh and Gera (2008)) and studied this problem for many classes of graphs: paths, cycles, wheels, stars, complete graphs and complete k-partite graphs. Moreover, we prove that this problem is NP-complete for graphs with a degree bounded by 4 and state an open question regarding subcubic graphs.