Computing the treewidth and the minimum fill-in with the modular decomposition

Abstract

Using the notion of modular decomposition we extend the class of graphs on which both the treewidth and the minimum fill-in problems can be solved in polynomial time. We show that if C is a class of graphs which is modularly decomposable into graphs that have a polynomial number of minimal separators, or graphs formed by adding a matching between two cliques, then both the treewidth and the minimum fill-in problems on C can be solved in polynomial time. For the graphs that are modular decomposable into cycles we give algorithms, that use respectively O(n) and O(n3) time for treewidth and minimum fill-in.