Locally chordal graphs

TL;DR

This paper explores locally chordal graphs, providing four new characterizations using induced subgraphs, local separators, coverings, and binary cycle spaces, thereby advancing the understanding of their structural properties.

Contribution

It introduces four novel characterizations of locally chordal graphs, connecting local properties with global structure and expanding the theoretical framework.

Findings

Four characterizations of locally chordal graphs are established.

Local separators and coverings are effective tools for graph characterization.

Binary cycle space analysis offers a new perspective on chordal graphs.

Abstract

In this paper we study locally chordal graphs, i.e. graphs where every small-radius ball is chordal. We prove four characterizations of locally chordal graphs. Two are counterparts of the classic descriptions of chordal graphs via induced subgraphs and via minimal separators. For the latter, we rely on the local separators introduced in [CJKK25]. Another characterization is via the local covering, which was introduced in [DJKK22] to study local-global characteristics of graphs using coverings from topology. Our final characterization of locally chordal graphs is in terms of their binary cycle spaces. This gives a new characterization of chordal graphs as wheel-free graphs whose binary cycle space is generated by triangles. Together, these results demonstrate the potential of local-global tools to uncover rich new properties. Our results in this paper also form the basis of our local-global analysis of locally chordal graphs [AKb], where we develop a local-global perspective into structural characterizations.