Locally chordal graphs
Tara Abrishami, Paul Knappe, Jonas Kobler
TL;DR
This paper introduces the notion of $r$-locally chordal graphs and develops four equivalent characterizations linking local structure to global chordality via induced-subgraph descriptions, the $r$-local cover, algebraic cycle-space properties, and minimal local separators. It shows that $r$-locally chordal graphs are exactly the $r$-chordal wheel-free graphs, that their $r$-local covers are chordal, and that every minimal $r$-local separator is a clique, thereby connecting local and global viewpoints. An algebraic perspective via the binary cycle space reveals that in wheel-free graphs, $r$-chordality is equivalent to generation of short cycles by triangles, yielding a new chordality criterion. These results establish a robust local-global framework for analyzing locally chordal graphs and motivate further exploration of local-global phenomena in graph theory.
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.