A generalization of chordal graphs and the maximum clique problem

A graph is chordal or triangulated if it has no chordless cycle with four or more vertices. Chordal graphs are well known for their combinatorial and algorithmic properties. Here we introduce a generalization of chordal graphs, namely CSG(k) graphs. informally, a CSG(0) graph is a complete graph, and for k > 0, the class of CSG(k) graphs is defined inductively in a such manner that CSG(1) Graphs are chordal graphs. We show that CSG(k) Graphs inherit of the same kind of properties as chordal graph. As a consequence, we show that the maximum clique problem is polynomial on CSG(k) graphs while this problem is NP-hard in the general case. (C) 1997 Elsevier Science B.V.