Simplicial vertex

In graph theory, a simplicial vertex is a vertex whose closed neighborhood in a graph forms a clique, where every pair of neighbors is adjacent to each other.
A vertex of a graph is bisimplicial if the set of it and its neighbours is the union of two cliques, and is -simplicial if the set is the union of cliques. A vertex is co-simplicial if its non-neighbours form an independent set.
Addario-Berry et al. demonstrated that every even-hole-free graph contains a bisimplicial vertex, which settled a conjecture by Reed. The proof was later shown to be flawed by Chudnovsky & Seymour, who gave a correct proof. Due to this property, the family of all even-cycle-free graphs is -bounded.