Conflict-Free Cuts: Exploring the Boundaries of Tractability

Abstract

A conflict-free cut of a host graph $G$ is an edge cut containing no conflicting pair of edges. The conflicts are specified by a conflict graph $\widehat G$ whose vertex set is $E(G)$. The conflict-free cut problem asks if the given instance $(G, \widehat G)$ has a conflict-free cut. When restricted to planar $G$ and 1-regular $\widehat G$, the problem is polynomial time solvable when $G$ has maximum degree 4, but NP-complete already at maximum degree 5. We study the complexity of conflict-free cut problem under restrictions on the degrees of $G$, the components of $\widehat G$, and the locality of the conflicts. A conflict is local if its two edges share an endpoint in $G$. We prove the following results. * For $Δ(\widehat G)\le1$, a directed-cycle formulation gives a polynomial-time algorithm when $|E(G)|\le2|V(G)|$, hence for maximum degree four. The problem is NP-complete on $5$-regular hosts with exactly $1$-regular conflicts. * For $Δ(\widehat G)\le1$, we give a constructive FPT algorithm in $s_4(G)=\sum_v\max\{0,d_G(v)-4\}$, single-exponential for planar hosts. For arbitrary conflicts, we give an XP algorithm in $η=\max\{0,|E(\widehat G)|-|V(G)|\}$. * On $4$-regular hosts, local exactly $2$-regular conflicts with components $C_3,C_4$ give NP-completeness. For local conflicts at maximum degree four, pair or triangle components separately permit polynomial algorithms, but mixing them gives NP-completeness. We classify cases where $\widehat G$ is a union of cliques. * An $a$-COC is a vertex set whose deletion leaves components of order at most $a$. For local conflicts, the problem is FPT in the minimum $2$-COC size, but NP-complete with a supplied $3$-COC of size six. Without locality, hardness holds with a supplied $2$-COC of size six. Both hardness results use exactly $1$-regular conflicts.