A Characterization of all Stable Minimal Separator Graphs

TL;DR

This paper characterizes graphs where all minimal vertex separators are stable sets, showing they exclude a specific cycle with a chord, and proves that graphs with minimal edge separators inducing matchings are exactly trees.

Contribution

It provides a structural characterization of graphs with stable minimal vertex separators and establishes NP-completeness for recognizing certain forbidden subgraphs.

Findings

Graphs with all minimal vertex separators stable exclude a cycle with one chord.

Deciding the maximum such forbidden subgraph is NP-complete.

Graphs where all minimal edge separators induce matchings are exactly trees.

Abstract

In this paper, our goal is to characterize two graph classes based on the properties of minimal vertex (edge) separators. We first present a structural characterization of graphs in which every minimal vertex separator is a stable set. We show that such graphs are precisely those in which the induced subgraph, namely, a cycle with exactly one chord is forbidden. We also show that deciding maximum such forbidden subgraph is NP-complete by establishing a polynomial time reduction from maximum induced cycle problem [1]. This result is of independent interest and can be used in other combinatorial problems. Secondly, we prove that a graph has the following property: every minimal edge separator induces a matching (that is no two edges share a vertex in common) if and only if it is a tree.