Two classes of β-perfect graphs that do not necessarily have simplicial extremes

For a graph G, β(G) = max{δ(G′)+1 | G′ is an induced subgraph of G}. This parameter is an upper bound on the chromatic number of a graph. A graph G is β-perfect if χ(G′) = β(G′) for all induced subgraphs G′ of G. A number of classes have been shown in literature to be β-perfect, but for the class of all β-perfect graphs, the complexity of their recognition and characterization in terms of forbidden induced subgraphs remain open.

It is known that minimally β-imperfect graphs cannot have a simplicial extreme, i.e. a vertex whose neighborhood is a clique or of size 2. β-perfection of all the known β-perfect classes of graphs was shown through the existence of simplicial extremes. In this paper we study two classes of β-perfect graphs that do not necessarily have this property.

A hole is a chordless cycle of length at least 4, and it is even or odd depending on the parity of its length. β-perfect graphs cannot contain even holes as induced subgraphs. We prove that graphs that do not contain an even hole, a twin wheel nor a cap as an induced subgraph are β-perfect (where a twin wheel is a graph that consists of a hole and a vertex that has exactly 3 neighbors on the hole, that are furthermore consecutive on the hole; and a cap is a graph that consist of a hole and a vertex that has exactly 2 neighbors on the hole, that are furthermore consecutive on the hole). This class properly contains chordal graphs, and is the only known generalization of chordal graphs that is shown to be β-perfect.

A hyperhole is a graph that is obtained from a hole by a sequence of clique substitutions. We give a complete structural characterization of β-perfect hyperholes, which we then use to give a linear-time algorithm to recognize whether a hyperhole is β-perfect. We also obtain a complete list of minimally β-imperfect hyperholes.