Minimal separators in P4P4-sparse graphs

In this paper, we determine the minimal separators of P4P4-sparse graphs and establish bounds on their number. Specifically, we show that a P4P4-sparse graph G on n vertices and m   edges has fewer than 2n/32n/3 minimal separators of total description size at most 4m/34m/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 P4P4-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 P4P4-sparse graph; the algorithm can be modified to report the minimal separators of the input graph in linear time and space as well.