A defensive alliance in an undirected graph $G=(V,E)$ is a non-empty set of vertices $S$ satisfying the condition that every vertex $v\in S$ has at least as many neighbours (including itself) in $S$ as it has in $V\setminus S$. We consider the notion of global minimality in this paper. We are interested in globally 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 by showing that the Globally Minimal Defensive Alliance problem is FPT parameterized by the neighbourhood diversity of the input graph. The result for neighborhood diversity implies that the problem is FPT parameterized by vertex cover number also. We prove that the problem parameterized by the vertex cover number of the input graph does not admit a polynomial compression unless coNP $\subseteq$ NP/poly. We show that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, pathwidth, treewidth and treedepth. We also proved that, given a vertex $r \in V(G)$, deciding if $G$ has a globally minimal defensive alliance of any size containing vertex $r$ is NP-complete.
翻译:在未方向的图形$G=(V,E)中,一个防御性联盟$G=(V,E)美元是一个非空的顶点组合,它满足了一个条件,即每个顶端防守联盟问题都比输入图的相邻方多,而每个顶端防守联盟问题是FPT的参数。我们在本文件中考虑了全球最小度概念。我们感兴趣的是最大尺寸的全球最低防御性联盟。这个问题已知是NP-硬的,但其参数复杂性一直到现在才开放。我们从参数化复杂度的观点出发,加深了我们对问题的了解,显示全球最小防守联盟问题是由输入图的相邻方多样性参数化的FPT参数(包括它本身)。 社区多样性的结果还表明,问题是以顶点覆盖数的参数化为全球最小值。我们证明,由顶点覆盖的输入图数所标定的问题并不包含多度压缩的压缩值压缩值,除非CONP=subset $NPP/poly。我们显示,问题是由一系列相当限制性的结构型的G-wial确定最起码的一条路径,例如反馈。