A stable cutset is a set of vertices $S$ of a connected graph, that is pairwise non-adjacent and when deleting $S$, the graph becomes disconnected. Determining the existence of a stable cutset in a graph is known to be NP-complete. In this paper, we introduce a new exact algorithm for Stable Cutset. By branching on graph configurations and using the $O^*(1.3645)$ algorithm for the (3,2)-Constraint Satisfaction Problem presented by Beigel and Eppstein, we achieve an improved running time of $O^*(1.2972^n)$. In addition, we investigate the Stable Cutset problem for graphs with a bound on the minimum degree $\delta$. First, we show that if the minimum degree of a graph $G$ is at least $\frac{2}{3}(n-1)$, then $G$ does not contain a stable cutset. Furthermore, we provide a polynomial-time algorithm for graphs where $\delta \geq \tfrac{1}{2}n$, and a similar kernelisation algorithm for graphs where $\delta = \tfrac{1}{2}n - k$. Finally, we prove that Stable Cutset remains NP-complete for graphs with minimum degree $c$, where $c > 1$. We design an exact algorithm for this problem that runs in $O^*(\lambda^n)$ time, where $\lambda$ is the positive root of $x^{\delta + 2} - x^{\delta + 1} + 6$. This algorithm can also be applied to the \textsc{3-Colouring} problem with the same minimum degree constraint, leading to an improved exact algorithm as well.

翻译：稳定割集是指连通图中一个两两不相邻的顶点集合$S$，当删除$S$后图变得不连通。已知判定图中是否存在稳定割集是NP完全问题。本文针对稳定割集问题提出了一种新的精确算法。通过对图构型进行分支，并利用Beigel和Eppstein提出的(3,2)-约束满足问题的$O^*(1.3645)$算法，我们实现了$O^*(1.2972^n)$的改进运行时间。此外，我们研究了具有最小度$\delta$约束的图的稳定割集问题。首先证明若图$G$的最小度至少为$\frac{2}{3}(n-1)$，则$G$不包含稳定割集。进一步地，我们针对$\delta \geq \tfrac{1}{2}n$的图给出了多项式时间算法，并针对$\delta = \tfrac{1}{2}n - k$的图设计了类似的核化算法。最后，我们证明了对于最小度为$c$（其中$c > 1$）的图，稳定割集问题仍保持NP完全性。为此我们设计了一个精确算法，其运行时间为$O^*(\lambda^n)$，其中$\lambda$是方程$x^{\delta + 2} - x^{\delta + 1} + 6$的正根。该算法同样适用于具有相同最小度约束的\textsc{3-着色}问题，从而也得到了改进的精确算法。