The Multi-terminal Vertex Separator Problem: Polytope Characterization and TDI-ness

Abstract

In this paper we discuss a variant of the well-known k-separator problem. Consider the simple graph \(G=(V\cup T,E)\) with \(V\cup T\) the set of vertices, where T is a set of distinguished vertices called terminals, inducing a stable set and E a set of edges. Given a weight function \(w: V\rightarrow \mathbb N\), the multi-terminal vertex separator problem consists in finding a subset \(S\subseteq V\) of minimum weight intersecting every path between two terminals. We characterize the convex hull of the solutions of this problem in two classes of graph which we call, star trees and clique stars. We also give TDI systems for the problem in these graphs.