On Generalized Perfect Graphs: Bounded Degree and Bounded Edge Perfection

Given a hereditary family of graphs P one defines the P-chromatic number of a graph G, denoted χP(G), to be the minimum size of a partition V(G)=V1∪⋯∪Vk such that each Vi induces in G a member of P. Define ωP(G) to equal χP(K) where K is a largest clique in G. We say that G is χP-perfect provided χP(H)=ωP(H) for all induced subgraphs H of G. We consider the properties Et, “has at most t edges” and Dt, “has maximum degree at most t”. For these properties (and some variants) we prove analogues of the Strong Perfect Graph Conjecture and the Perfect Graph Theorem, and we also exhibit polynomial time algorithms for recognizing these generalized perfect graphs provided t>0.