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既是独立集又是割集的顶点集。即使输入图为平面图且最大度数为5，该问题仍然是$\textsf{NP}$-完全的。本文首先提出一个$\mathcal{O}^*(1.4423^{n})$时间复杂度的算法来解决该问题。我们还展示了如何在相同运行时间内计算最小独立割集（如果存在）。由于具有独立割集的性质是MSO$_1$可表达的这一特性，我们的主要结果关注该问题在参数不被输入图团宽函数所界定时的结构参数化。我们针对以下参数提出了$\textsf{FPT}$时间复杂度的算法：最大度数的对偶、解大小的对偶、支配集的大小（其中支配集作为额外输入给出）、奇环横贯的大小、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，这使我们能够识别更多固定参数可解和多时间可解的情况。