Canonical tree-decompositions of chordal graphs

TL;DR

This paper characterizes locally finite, connected graphs that are locally chordal using a canonical tree-decomposition into cliques, linking local properties to a global structural decomposition.

Contribution

It introduces a canonical version of Halin's characterization, ensuring the tree-decomposition into cliques is invariant under automorphisms.

Findings

A graph is r-locally chordal iff its canonical r-global structure decomposes into cliques.

Canonical tree-decompositions can be chosen to be automorphism-invariant.

The proof connects local chordality with a global clique-based decomposition.

Abstract

We show that a locally finite, connected graph $G$ is $r$-locally chordal (that is, its $r/2$-balls are chordal) if and only if the unique canonical graph-decomposition $\mathcal{H}_r(G)$ of $G$ displaying its $r$-global structure is into cliques. Our proof relies on a canonical version of Halin's characterization of chordal locally finite graphs as those that admit a tree-decomposition into cliques: We show that such tree-decompositions can be chosen to be canonical, that is, so that they are invariant under all the graph's automorphisms.