Optimizing slightly triangulated graphs

Graphs is called slightly if it contaiw; no chordless cyck or more vertices and induced subgrapb has a vertex whose nellgnLoc1urnooCl contains no induced path on four verticf~S. These graphs triangulated graphs and appear naturally in the study of thf~ inhcrsection graphs of the maximal rectangles of trii::.tngulated graphs are perfect (in the sense of Berge). In this paper WI? present algorithms for slightly triangulated graphs, for ('l, maXImum and for finding an optimal colouring. Let Pk denote the chordless path with k vertices, Ck the chordless with k vertices and G the complement of the graph G. We write H c G if H is an induced subgraph of G. The maximum size of a clique in G is denoted by w(G), and the number of vertices in G is denoted by n. The neighbourhood of a vertex x in a su bgraph H is denoted by r H (x). In the early sixties Berge [1] defined a graph G to be p(~rfect if for every induced subgraph H of G the chromatic number of H is equal to the largest size of a clique in H. Graphs which played an important role in the development of perfect graph theory are the triangulated graphs. A graph is triangulated if it contains no chordless cycle with four or more vertices. Another characterization is that every induced subgraph of a triangulated graph contains a vertex whose neighbourhood is a clique. Triangulated graphs constitute a large class of perfect graphs with numerous applications. They have been thoroughly studied and efficient algorithms are known for these graphs. The reader is referred to [5] for an introduction to this topic. We introduced in [9] also [10]) a generalization of triangulated graphs; we call a graph 8lightly triangulated if it satisfies the following two conditions. 1. Vk 2 5, (\ ct G Australasian Journal of Combinatorics 1i (1996), pp. 5-13