Canonical tree-decompositions of chordal graphs

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.