Triangulated Neighbourhoods in C ₄-Free Berge Graphs

Abstract

We call a T-vertex of a graph G = (V,E) a vertex z whose neighbourhood N(z) in G induces a triangulated graph, and we show that every C ₄-free Berge graph either is a clique or contains at least two non-adjacent T-vertices. An easy consequence of this result is that every C ₄-free Berge graph admits a T-elimination scheme, i.e. an ordering [v ₁, v ₂, . . . , v _n] of its vertices such that vi is a T-vertex in the subgraph induced by v _i, . . . , v _n in G.