Canonical tree-decompositions of chordal graphs
Raphael W. Jacobs, Paul Knappe
TL;DR
This work establishes that locally finite, connected chordal graphs admit a canonical tree-decomposition into cliques, invariant under automorphisms, and strengthens this to maximal-clique decompositions in the presence of normal coverings. It connects canonical clique-based decompositions with r-local coverings to characterize r-locally chordal graphs via their Hr-structures, showing the r-global structure can be read off from a canonical decomposition into (maximal) cliques. The authors also present sharp counterexamples illustrating that strengthening to maximal cliques or achieving canonicity in broader settings is not always possible, and they discuss how coverings can recover canonical maximal-clique decompositions in the locally chordal context. Overall, the paper advances a local-to-global perspective on chordal and locally chordal graphs through canonical graph-decompositions and coverings.
Abstract
Halin characterised the chordal locally finite graphs as those that admit a tree-decomposition into cliques. We show that these tree-decompositions can be chosen to be canonical, that is, so that they are invariant under all the graph's automorphisms. As an application, 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 $G$ which displays its $r$-global structure is into cliques. Our results also serve as tools for further characterisations of $r$-locally chordal graphs.