Minimal separators in [formula omitted]-sparse graphs

Abstract

In this paper, we determine the minimal separators of P 4 -sparse graphs and establish bounds on their number. Specifically, we show that a P 4 -sparse graph G on n vertices and m edges has fewer than 2 n / 3 minimal separators of total description size at most 4 m / 3 . The bound on the number of minimal separators is tight and is also tight for the class of cographs, a well known subclass of the P 4 -sparse graphs. Our results enable us to present a linear-time and linear-space algorithm for computing the number of minimal separators of a given P 4 -sparse graph; the algorithm can be modified to report the minimal separators of the input graph in linear time and space as well.