Graph classes with and without powers of bounded clique-width

Abstract

We initiate the study of graph classes of power-bounded clique-width, that is, graph classes for which there exist integers k and ℓ such that the kth powers of the graphs are of cliquewidth at most ℓ. We give sufficient and necessary conditions for this property. As our main results, we characterize graph classes of power-bounded clique-width within classes defined by either one forbidden induced subgraph, or by two connected forbidden induced subgraphs. We also show that for every positive integer k, there exists a graph class such that the kth powers of graphs in the class form a class of bounded clique-width, while this is not the case for any smaller power.