k-Cographs are Kruskalian

A class of graphs is Kruskalian if Kruskal's theorem on a well-quasi-ordering of finite trees provides a finite characterization in terms of forbidden induced subgraphs. Let k be a natural number. A graph is a k-cograph if its vertices can be colored with colors from the setf1;:::; kg such that for every nontrivial subset of vertices W there exists a partition fW1;W2g of W into nonempty subsets such that either no vertex of W1 is adjacent to a vertex of W2 or, such that there exists a permutation p2 Sk such that vertices with color i in W1 are adjacent exactly to the vertices with color p(i) in W2, for all i2f1;:::; kg. We prove that k-cographs are Kruskalian. We show that k-cographs have bounded rankwidth and that they can be recognized in O(n 3 ) time.