Closed graphs are proper interval graphs

TL;DR

This paper proves that all closed graphs are equivalent to proper interval graphs, leading to efficient linear-time algorithms for recognizing such graphs.

Contribution

It establishes the isomorphism between closed graphs and proper interval graphs, providing a new characterization and practical recognition algorithms.

Findings

Closed graphs are isomorphic to proper interval graphs

Linear-time algorithms exist for recognizing closed graphs

Enhanced understanding of graph classes and their properties

Abstract

In this note we prove that every closed graph $G$ is up to isomorphism a proper interval graph. As a consequence we obtain that there exist linear-time algorithms for closed graph recognition.