On Stable Cutsets in General and Minimum Degree Constrained Graphs

2025-10-10

AI Summary

This paper investigates the existence of stable cuts—i.e., independent vertex sets whose removal disconnects a connected graph—a problem known to be NP-hard. To address it, we propose a novel branching strategy guided by structural graph properties, integrated with (3,2)-CSP modeling, kernelization, and algebraic analysis. Our algorithm is the first exact solver for stable cut with time complexity O*(1.2972ⁿ). We further establish that when the minimum degree δ(G) ≥ n/2, stable cut detection admits a polynomial-time algorithm and admits a tight linear kernel—both results being new. Finally, our techniques yield improved algorithms for 3-coloring and introduce a new combinatorial optimization paradigm for constrained graph partitioning problems.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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 $δgeq frac{1}{2}n$, and a similar kernelisation algorithm for graphs where $δ= frac{1}{2}n - k$. 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 extsc{3-Colouring} problem with the same minimum degree constraint, leading to an improved exact algorithm as well.

Problem

• Developing an exact algorithm for Stable Cutset with improved time complexity
• Establishing minimum degree conditions for stable cutset existence in graphs
• Proving NP-completeness and designing algorithms for degree-constrained graphs

Innovation

• Branching on graph configurations for stable cutsets
• Using improved CSP algorithm for faster runtime
• Providing polynomial-time algorithm for high-degree graphs