Efficient Algorithms for Minimum Weighted Colouring of Some Classes of Perfect Graphs

We design an O(m) algorithm to find a minimum weighted colouring and a maximum weighted clique of a perfectly ordered graph. We also present two O(n’) algorithms to find a minimum weighted colouring of a comparability graph and of a triangulated graph. Our colouring algorithms use an algorithm to find a stable set meeting all maximal (with respect to set inclusion) cliques of a perfectly ordered graph. We show that the problem of finding such stable set in an arbitrary graph is NP-hard. We shall also describe a polynomial algorithm to find a minimum weighted colouring of a clique separable graph.