A Separator Theorem for Chordal Graphs

Abstract

Chordal graphs are undirected graphs in which every cycle of length at least four has a chord. They are sometimes called triangulated graphs, monotone transitive graphs, rigid circuit graphs, or perfect elimination graphs; the last name reflects their utility in modelling Gaussian elimination on sparse matrices. The main result of this paper is that a chordal graph with n vertices and m edges can be cut in half by removing $O(\sqrt m )$ vertices. A similar result holds if the vertices have nonnegative weights and we want to bisect the graph by weight, or even if we want to bisect the graph simultaneously by several unrelated sets of weights. We present an $O(m)$ time algorithm to find the separating set.