Generalized perfect graphs: Characterizations and inversion

Abstract

Given a hereditary family of graphs P one defines the P -chromatic number of a graph G (denoted χ P (G)) to be the manimum size of a partition V( G) = V 1 ∪ 3· ∪ V k such that each V i induces in G a member of P . Define ω P (G) to equal max {χ P (K)} where the maximum is taken over all cliques K in G. We say that G is χ P - perfect provided χ P (H) = ω P (H) for all induced subgraphs H of G and we denote the set of χ P - perfect graphs by P ∗. In this paper we discuss the following results: 1. (1) We give analogs of the Strong Perfect Graph Conjecture, that is, we find forbidden subgraph characterizations of P ∗ for various families P . 2. (2) We show the central role played by the classes Free( K n ) = { G: ω( G) < n} in finding P ∗ for all hereditary P , and give a partial characterization of ( Free(K n)) ∗ for n ⩾ 3 . 3. (3) We consider the problem of inverting perfection: given a hereditary family Q , find all hereditary P such that P ∗ = Q . We find conditions on P that are necessary and sufficient for P ∗ = Q . We then apply this “inverting perfection theorem” to a number of families Q .