Offensive alliances in graphs

Abstract

The Offensive Alliance problem has been studied extensively during the last twenty years. A set S⊆V of vertices is an offensive alliance in an undirected graph G=(V,E) if each v∈N(S) has at least as many neighbors in S as it has neighbors (including itself) not in S. We study the classical and parameterized complexity of the Offensive Alliance problem, where the aim is to find a minimum size offensive alliance. We enhance our understanding of the problem from the viewpoint of parameterized complexity by showing that (1) the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, treewidth, pathwidth, and treedepth of the input graph; this puts it among the few problems that are FPT when parameterized by solution size but not when parameterized by treewidth (unless FPT=W[1]), (2) the problem cannot be solved in time O⁎(2o(klog⁡k)) where k is the solution size, unless ETH fails, (3) it does not admit a polynomial kernel parameterized by solution size and vertex cover of the input graph. On the positive side we prove that (4) it can be solved in time O⁎(vc(G)O(vc(G))) where vc(G) is the vertex cover number of the input graph. (5) it admits an FPT algorithm when parameterized by vertex integrity of input graph. In terms of classical complexity, we prove that (6) the problem cannot be solved in time 2o(n) even when restricted to bipartite graphs, unless ETH fails, (7) it cannot be solved in time 2o(n) even when restricted to apex graphs, unless ETH fails. We also prove that (8) it is NP-complete even when restricted to bipartite, chordal, split and circle graphs.