On the Clique—Width of Perfect Graph Classes

Abstract

Graphs of clique-width at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs which can be defined by k-expressions based on graph operations which use k vertex labels. In this paper we study the clique-width of perfect graph classes. On one hand, we show that every distance-hereditary graph, has clique-width at most 3, and a 3-expression defining it can be obtained in linear time. On the other hand, we show that the classes of unit interval and permutation graphs are not of bounded clique-width. More precisely, w e show that for every n ∈ N there is a unit interval graph In and a permutation graph Hn having n2 vertices, each of whose clique-width is exactly n+1. These results allowus to see the borderwithin the hierarchy of perfect graphs between classes whose clique-width is bounded and classes whose clique-width is unbounded. Finally we show that every n×n square grid, n ∈ N, n ≥ 3, has clique-width exactly n + 1.