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合。即使输入图为平面图且最大度为五，该问题仍然是$\textsf{NP}$-完全的。本文首先提出一种复杂度为$\mathcal{O}^*(1.4423^{n})$的算法来解决该问题，并展示了如何在相同运行时间内计算出最小独立割集（若存在）。由于存在独立割集的性质是MSO$_1$可表达的，我们的主要结果关注该问题针对某些参数的参数化算法，这些参数不受输入图的团宽的函数限制。我们针对以下参数提出了$\textsf{FPT}$-时间算法：最大度的对偶、解规模的对偶、支配集的大小（其中支配集作为附加输入给出）、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，这使我们能够识别出更多的固定参数可解情形和多项式时间可解情形。