Parameterized Complexity of Locally Minimal Defensive Alliances
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The Defensive Alliance problem has been studied extensively during the last twenty years. A set S of vertices of a graph is a defensive alliance if, for each element of S, the majority of its neighbours is in S. We consider the notion of local minimality in this paper. We are interested in locally minimal defensive alliance of maximum size. This problem is known to be NP-hard but its parameterized complexity remains open until now. We enhance our understanding of the problem from the viewpoint of parameterized complexity. The main results of the paper are the following: (1) when the input graph happens to be a tree, Connected Locally Minimal Strong Defensive Alliance can be solved in polynomial time, (2) the Locally Minimal Defensive Alliance problem is NP-complete, even when restricted to planar graphs, (3) a color coding algorithm for Exact Connected Locally Minimal Defensive Alliance, (4) the Locally Minimal Defensive Alliance problem is fixed parameter tractable (FPT) when parametrized by neighbourhood diversity, (5) the Exact Connected Locally Minimal Defensive Alliance problem parameterized by treewidth is W[1]-hard and thus not FPT (unless FPT=W[1]), (6) Locally Minimal Defensive Alliance can be solved in polynomial time for graphs of bounded treewidth.
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