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.

中文翻译：

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局部最小防御联盟的参数化复杂性

在过去的20年中，防御性联盟问题得到了广泛的研究。如果对于图元的每个元素，大多数邻居在图元中，则图的一组顶点的元图元是一个防御性联盟。我们在本文中考虑局部极小值的概念。我们对最大规模的本地最小防御联盟感兴趣。已知此问题是NP难题，但其参数化复杂性至今仍未解决。从参数化复杂性的角度来看，我们增强了对问题的理解。本文的主要结果如下：（1）当输入图碰巧是一棵树时，可以在多项式时间内求解局部最小防御强联盟[2]；局部最小防御联盟问题是NP完全，即使仅限于平面图，