Upper bounds to the clique width of graphs

A graph complexity measure that we call clique-width is associated in a natural way with certain graph decompositions, more or less like tree-width is associated with tree-decomposition which are, actually, hierarchical decompositions of graphs. In general, a decomposition of a graph G can be viewed as a nite term, written with appropriate operations on graphs, that evaluates to G. Innnitely many operations are necessary to deene all graphs. By limiting the operations in terms of some integer parameter k, one obtains complexity measures of graphs. Speciically, a graph G has complexity at most k ii it has a decomposition deened in terms of k operations. Hierarchical graph decompositions are interesting for algorithmic purposes. In fact, many NP-complete problems have linear algorithms on graphs of tree-width or of clique-width bounded by some xed k, and the same will hold for graphs of clique-width at most k. The graph operations upon which clique-width and the related decompositions are based have been already introduced by the rst author in relation with the description of certain context-free graph grammars in terms of systems of mutually recursive equations. These operations build graphs as gluings of complete bipartite graphs. We call clique-width the corresponding graph complexity measure to recall this aspect { cliques are not far from complete bipartite graphs. Clique-width is a diicult notion. We do not even know whether graphs of clique-width at most 3 are recognizable in polynomial time. Our approach here is twofold: rst, we investigate classes of graphs that have uniformly bounded clique-width, in particular, graphs of bounded tree-width; second we deene graph transformations that leave unchanged the clique-width or increase it in a controlled way.