On the Parameterized Complexity of Conflict-Free Edge Cut in Undirected Graphs

2026-10-05

AI Summary

This study addresses the parameterized complexity and polynomial kernelization challenges of the Conflict-Free Edge Cut (CF-CUT) problem and its minimization variant (MIN-CF-CUT). By employing parameterized complexity theory and fixed-parameter tractability analysis, this work introduces, for the first time, the dissociation number to characterize how structural graph parameters influence computational hardness. The contributions systematically establish the algorithmic feasibility boundaries of CF-CUT, revealing its NP-hardness and polynomial-time solvability under specific conditions. Furthermore, it proves that MIN-CF-CUT is W[1]-hard under multiple parameterizations and rules out the existence of polynomial kernels, thereby clarifying the kernelization barriers and theoretical limits of this problem.

Abstract

In this paper, we study CONFLICT-FREE EDGE CUT (CF-CUT), which is a recently introduced conflict-free version of the MIN-CUT problem that asks to find the minimum number of edges to disconnect a connected graph. The CF-CUT takes as input a connected undirected graph G = (V, E), a conflict graph $\widehat{G}$ such that $E(G) = V(\widehat{G})$, and the objective is to decide whether there exists $F \subseteq E(G)$ such that $G - F$ is disconnected and $F$ is an independent set in $\widehat{G}$. Rauch et. al. [IPL-2025] proved that CF-CUT is NP-Complete and also provided some results on the parameterized complexity of CF-CUT. A related variant MIN CONFLICT-FREE EDGE CUT (MIN-CF-CUT) takes a connected graph $G$, a conflict graph $\widehat{G}$ such that $V(\widehat{G}) = E(G)$, and an integer $k$ as input and asks if there is a set $F$ of at most $k$ edges such that $G - F$ is disconnected and $F$ is an independent set in $\widehat{G}$. In this paper, we extend the work of Rauch et al. [IPL-2025] and provide a systematic study on the CF-CUT and MIN-CF-CUT from the perspective of parameterized complexity and polynomial kernelization. We prove that CF-CUT is NP-hard when the dissociation number of the input graph is at most two. We also complement it by proving that CF-CUT is poly-time solvable when the dissociation number of an input graph is at most one. Additionally, for MIN-CF-CUT, we consider both solution size and various structural parameters of the input graph as parameters, and provide fixed-parameter tractability and W[1]-hardness results when the conflict graph is restricted to various graph classes. We also prove that unless NP $\subseteq$ coNP/poly, MIN-CF-CUT admits no polynomial kernel when parameterized by the vertex cover number of the input graph; and also when parameterized by the sum of the solution size and the vertex integrity of the input graph.

Problem

• Conflict-Free Edge Cut
• Parameterized Complexity
• Polynomial Kernelization
• Fixed-Parameter Tractability
• Undirected Graphs

Innovation

• Conflict-Free Edge Cut
• Parameterized Complexity
• Polynomial Kernelization
• Dissociation Number
• Fixed-Parameter Tractability