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

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.