Locally interval graphs are circular-arc graphs

TL;DR

This paper proves that all locally interval graphs are also circular-arc graphs, establishing a key equivalence in graph theory related to local and global intersection properties.

Contribution

It demonstrates that the class of locally interval graphs coincides exactly with circular-arc graphs, filling a gap in the understanding of their structural relationship.

Findings

Every locally interval graph is a circular-arc graph

The result connects local properties to global graph classes

Builds on previous work on locally chordal graphs

Abstract

Circular-arc graphs are graphs that can be represented as intersection graphs of subpaths of a cycle. Interval graphs are graphs that can be represented as intersection graphs of subpaths of a path. Since cycles are locally paths, every circular-arc graph is locally interval. In this paper, we prove that the converse holds as well: every locally interval graph is a circular-arc graph. This result and its proofs are connected to a recent broader study of structural local-global theory and build on previous work on locally chordal graphs.