On simplicial and co-simplicial vertices in graphs

We 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.