The Independent Cutset problem asks whether there is a set of vertices in a given graph that is both independent and a cutset. Such a problem is $\textsf{NP}$-complete even when the input graph is planar and has maximum degree five. In this paper, we first present a $\mathcal{O}^*(1.4423^{n})$-time algorithm for the problem. We also show how to compute a minimum independent cutset (if any) in the same running time. Since the property of having an independent cutset is MSO$_1$-expressible, our main results are concerned with structural parameterizations for the problem considering parameters that are not bounded by a function of the clique-width of the input. We present $\textsf{FPT}$-time algorithms for the problem considering the following parameters: the dual of the maximum degree, the dual of the solution size, the size of a dominating set (where a dominating set is given as an additional input), the size of an odd cycle transversal, the distance to chordal graphs, and the distance to $P_5$-free graphs. We close by introducing the notion of $\alpha$-domination, which allows us to identify more fixed-parameter tractable and polynomial-time solvable cases.

翻译：独立割集问题询问给定图中是否存在一个顶点集，该集合既是独立集又是割集。即使输入图是平面图且最大度为5，该问题仍然是$\textsf{NP}$-完全的。在本文中，我们首先提出了一种$\mathcal{O}^*(1.4423^{n})$时间复杂度的算法来解决该问题，并展示了如何在同一运行时间内计算出最小独立割集（如果存在）。由于具有独立割集的性质是MSO$_1$可表达的，我们的主要结果关注于该问题的结构参数化，考虑那些不受输入图团宽函数界定的参数。我们针对以下参数提出了$\textsf{FPT}$时间算法：最大度的对偶、解集大小的对偶、支配集的大小（其中支配集作为额外输入给出）、奇圈横贯的大小、到弦图的距离，以及到$P_5$-自由图的距离。最后，我们引入了$\alpha$-支配的概念，从而识别出更多固定参数可解和多式时间可解的情形。