On Stable Cutsets in General and Minimum Degree Constrained Graphs

TL;DR

The paper tackles the Stable Cutset problem, proving NP-completeness and delivering a new exact algorithm that runs in $O^*(1.2972^n)$ by modeling instances as annotated graphs and reducing to a $(3,2)$-CSP via Beigel–Eppstein techniques. It then explores how minimum-degree constraints influence stability, establishing a sharp upper bound $ abla > \frac{2}{3}(n-1)$ beyond which no stable cutset exists, providing a polynomial-time algorithm for $\delta \ge \frac{1}{2}n$, and a kernel for $\delta = \frac{1}{2}n - k$. The paper also proves NP-completeness for constant $c>1$ and presents an $O^*(\lambda^n)$-time exact algorithm with $\lambda$ the positive root of $x^{\delta+2}-x^{\delta+1}-6$, which also yields improvements for $3$-Colouring under the same minimum-degree constraint. Overall, the work ties together annotation-based modelling, CSP reductions, and measure-and-conquer analysis to advance exact algorithms for Stable Cutset and related colouring problems in graphs with degree constraints, with practical implications for fast exact methods in these domains.

Abstract

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 $δ$. 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 $δ\geq \tfrac{1}{2}n$, and a similar kernelisation algorithm for graphs where $δ= \tfrac{1}{2}n - k$. 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 $O^*(λ^n)$ time, where $λ$ is the positive root of $x^{δ+ 2} - x^{δ+ 1} + 6$. 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.

Figures (8)

• Figure 1: Example of some annotated graph $G$. Note, that $p(v_4) = \{A\}$ and therefore $B \notin p(v_2)$ and $B \notin p(v_3)$ by Lemma \ref{['lem:annGraphRules']}.
• Figure 2: The annotated graph from Figure \ref{['fig:annotatedgraphexample']} transformed to a (3,2)-CSP instance. The lines in the representation indicate the constraints.
• Figure 3: Branching Rule 1 and 2, respectively. The vertices highlighted in red correspond to the configurations being branched on, while the vertices in blue represent those that remain as standard variables in the (3,2)-CSP instance.
• Figure 4: Visualization of Case \ref{['case:exactAlgThreePerCorner']}. The vertices in orange correspond to the vertices in $r(T)$.
• Figure 5: Case \ref{['case:adjToTwoDiffVtxs']} visualized. The vertices in orange correspond to the vertices in $r(T_0)$.

Theorems & Definitions (27)

• Lemma 1
• Lemma 2
• Theorem 3
• Lemma 4: 3ColoringInTime1.3289^n
• Claim 5
• Claim 6
• Lemma 7
• Theorem 8
• Lemma 9
• Lemma 10