Bipartition of Graphs into Subgraphs with Prescribed Hereditary Properties

A hereditary class P is called finitely generated if the set of all minimal forbidden induced subgraphs for P is finite. For a pair of hereditary classes P and Q, we define a hereditary class P ∗ Q of all graphs G which have a partition A ∪ B = V (G) such that G(A) ∈ P and G(B) ∈ Q, where G(X) denotes the subgraph of G induced by X ⊆ V (G). We investigate the problem of recognizing finitely generated classes of the form P ∗Q. The following model is used. Let H and H be hypergraphs with the same vertex set V . The ordered pair H = (H,H) is called a bihypergraph. A bihypergraph H = (H,H) is called bipartite if there is an ordered partition V 0 ∪ V 1 = V (H) such that V i is stable in H for i = 0, 1. If the maximum cardinality of hyperedges in H is at most r and every k-subset of V (H) contains at least one hyperedge, then H ∈ C(k, r). It was proved in [4] that there exists a finite number of minimal non-bipartite bihypergraphs in C(k, r) whenever k and r are fixed. Let P and Q be hereditary classes of graphs. Suppose that the stability number α(H) is bounded above for all H ∈ P , and the clique number ω(H) is bounded above for all H ∈ Q. An ordered partition A∪B = V (G) is called a Ramseian P ∗Q-partition if G(A) ∈ P and G(B) ∈ Q. Let Ramsey(P ∗Q) be the set of all graphs having a Ramseian P ∗Q-partition. It follows from [4] that if both P and Q are finitely generated, then Ramsey(P ∗Q) is also finitely generated. In particular, every class of (α, β)-polar graphs generalizing split graphs has a finite forbidden induced subgraph characterization. We formulate a general conjecture that gives conditions for a class of graphs having a (P,Q)-partition to be finitely generated. New results supporting the conjecture are proved.