On the maximum weight minimal separator

Abstract

Given an undirected and connected graph G = (V, E) and two vertices s, t ∈ V, a vertex subset S that separates s and t is called an s-t separator, and an s-t separator is called minimal if no proper subset of S separates s and t. In this paper, we consider finding a minimal s-t separator with maximum weight on a vertex-weighted graph. We first prove that this problem is NP-hard. Then, we propose an twO(tw) n-time deterministic algorithm based on tree decompositions. Moreover, we also propose an O∗ (9tw W2)-time randomized algorithm to determine whether there exists a minimal s-t separator where W is its weight and tw is the treewidth of G.