Minimal vertex separators of chordal graphs

Abstract

Chordal graphs form an important and widely studied subclass of perfect graphs. The set of minimal vertex separators constitute an unique class of separators of a chordal graph and capture the structure of the graph. In this paper, we explore the connection between perfect elimination orderings of a chordal graph and its minimal vertex separators. Specifically, we prove a characterization of these separators in terms of the monotone adjacency sets of the vertices of the graph, numbered by the maximum cardinality search (MCS) scheme. This leads to a simple linear-time algorithm to identify the minimal vertex separators of a chordal graph using the MCS scheme. We also introduce the notion of multiplicity of a minimal vertex separator which indicates the number of different pairs of vertices separated by it. We prove a useful property of the lexicographic breadth first scheme (LBFS) that enables us to determine the multiplicities of minimal vertex separators of a chordal graph.