On simplicial and co-simplicial vertices in graphs

AbstractWe investigate the class of graphs defined by the property that every induced subgraph has a vertex which is either simplicial (its neighbours form a clique) or co-simplicial (its non-neighbours form an independent set). In particular we give the list of minimal forbidden subgraphs for the subclass of graphs whose vertex-set can be emptied out by first recursively eliminating simplicial vertices and then recursively eliminating co-simplicial vertices

On the disc-structure of perfect graphs II. The co-C4-structure

AbstractLet F be any family of graphs. Two graphs G1=(V1,E1),G2=(V2,E2) are said to have the same F-structure if there is a bijection f:V1→V2 such that a subset S induces a graph belonging to F in G1 iff its image f(S) induces a graph belonging to F in G2. We prove that if a C5-free graph H has the {2K2,C4}-structure of a perfect graph G then H is perfect