On the Clique-Width of Perfect Graph Classes

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, we show that for every n ∈ N there is a unit interval graph In and a permutation graph Hn having n 2 vertices, each of whose clique–width is exactly n+1. These results allow us to see the border within 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.