On Roussel-Rubio-type lemmas and their consequences

TL;DR

This paper presents a concise proof of a key lemma used in perfect graph theory and demonstrates how similar lemmas can simplify proofs of important theorems in weakly chordal and Meyniel graphs.

Contribution

It offers a new, shorter proof of the Roussel-Rubio lemma and applies it to streamline proofs of decomposition and structural results in graph theory.

Findings

Short proof of the main case of Roussel-Rubio lemma

Simplified proof of Hayward's decomposition theorem for weakly chordal graphs

Concise derivations of even pair existence in specific graph classes

Abstract

Roussel and Rubio proved a lemma which is essential in the proof of the Strong Perfect Graph Theorem. We give a new short proof of the main case of this lemma. In this note, we also give a short proof of Hayward's decomposition theorem for weakly chordal graphs, relying on a Roussel--Rubio-type lemma. We recall how Roussel--Rubio-type lemmas yield very short proofs of the existence of even pairs in weakly chordal graphs and Meyniel graphs.