Connectivity-Preserving Minimum Separator in AT-free Graphs

TL;DR

This paper introduces a polynomial-time algorithm for finding minimum-weight connectivity-preserving separators in AT-free graphs, addressing a problem that is NP-hard in general graphs and certain restricted classes.

Contribution

The paper proves that the safe $A,B$-separator problem is solvable in polynomial time specifically for AT-free graphs, contrasting with its NP-hardness in broader classes.

Findings

Polynomial-time algorithm for AT-free graphs

NP-hardness of the problem in general graphs

Existence of safe separators can be efficiently determined in AT-free graphs

Abstract

Let $A$ and $B$ be disjoint, non-adjacent vertex-sets in an undirected, connected graph $G$, whose vertices are associated with positive weights. We address the problem of identifying a minimum-weight subset of vertices $S\subseteq V(G)$ that, when removed, disconnects $A$ from $B$ while preserving the internal connectivity of both $A$ and $B$. We call such a subset of vertices a connectivity-preserving, or safe minimum $A,B$-separator. Deciding whether a safe $A,B$-separator exists is NP-hard by reduction from the 2-disjoint connected subgraphs problem, and remains NP-hard even for restricted graph classes that include planar graphs, and $P_\ell$-free graphs if $\ell\geq 5$. In this work, we show that if $G$ is AT-free then in polynomial time we can find a safe $A,B$-separator of minimum weight, or establish that no safe $A,B$-separator exists.