The Offensive Alliance problem has been studied extensively during the last twenty years. A set $S\subseteq V$ of vertices is an offensive alliance in an undirected graph $G=(V,E)$ if each $v\in N(S)$ has at least as many neighbours in $S$ as it has neighbours (including itself) not in $S$. We study the parameterized complexity of the Offensive Alliance problem, where the aim is to find a minimum size offensive alliance. Our focus here lies on parameters that measure the structural properties of the input instance. We enhance our understanding of the problem from the viewpoint of parameterized complexity by showing that 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.
翻译：在过去二十年中,对进攻性联盟问题进行了广泛研究。 一套固定的 $S\ subseteq Voctime of vertics 是非方向图形中的进攻性联盟 $G=(V,E)$,如果每美元都拥有至少与邻国(包括自己)没有的美元一样多的美元S值邻居。 我们研究了进攻性联盟问题的参数复杂性, 目的是找到最小规模的进攻性联盟。 我们这里的焦点在于测量输入实例结构属性的参数。 我们从参数化复杂度的角度来增进我们对问题的了解, 我们通过显示问题是由大量相当限制性的结构参数(例如输入图的反馈脊椎定序号、树边、路径宽度和树深度) 所设定的。
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