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Graphs of clique-width at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs which can be defined by fc-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 € Af there is a unit interval graph I n and a permutation graph H n having n 2 vertices, each of whose clique-width is at least n. 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 x n square grid, n £ JV/", n > 3, has clique-width exactly n + 1.

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We combine the known notion of the edge intersection graphs of paths in a tree with a VLSI grid layout model to introduce the edge intersection graphs of paths on a grid. Let P be a collection of nontrivial simple paths on a grid G. We define the edge intersection graph EPG(P) of P to have vertices which correspond to the members of P, such that two vertices are adjacent in EPG(P) if the corresponding paths in P share an edge in G. An undirected graph G is called an edge intersection graph of paths on a grid (EPG) if G = EPG(P) for some P and G, and P, G is an EPG representation of G. We prove that every graph is an EPG graph. A turn of a path at a grid point is called a bend. We consider here EPG representations in which every path has at most a single bend, called B 1 -EPG representations and the corresponding graphs are called B 1 -EPG graphs. We prove that any tree is a B 1 -EPG graph. Moreover, we give a structural property that enables one to generate non B 1 -EPG graphs. Furthermore, we characterize the representation of cliques and chordless 4-cycles in B 1 -EPG graphs. We also prove that single bend paths on a grid have Strong Helly number 3.