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 set {1, . . . ,k} such that for every nontrivial subset of vertices W there exists a partition {W1,W2} 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 π ∈ Sk such that vertices with color i in W1 are adjacent exactly to the vertices with color π(i) in W2, for all i ∈ {1, . . . ,k}. We prove that k-cographs are Kruskalian. We show that k-cographs have bounded rankwidth and that they can be recognized in O(n3) time.