Canonical tree-decompositions of chordal graphs

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.

💡 Research Summary

This paper investigates the existence of canonical tree‑decompositions for chordal graphs, i.e., tree‑decompositions that are invariant under the full automorphism group of the graph. Building on Halin’s classic characterisation—finite (or locally finite) chordal graphs are exactly those admitting a tree‑decomposition into maximal cliques—the authors ask whether such decompositions can be chosen to be canonical.

The first main result (Theorem 1) affirms this for locally finite, connected chordal graphs: every such graph admits a tree‑decomposition into cliques that is Aut$(G)$‑canonical. The proof proceeds by analysing minimal, tight, and efficient separators in chordal graphs. Using Dirac’s theorem that all minimal separators are cliques, together with a Helly‑type argument (Lemma 2.6) that any finite clique is contained in a single bag, the authors construct a decomposition recursively along minimal separators. Lemma 2.7 guarantees that any automorphism of $G$ induces at most one automorphism of the decomposition tree, which yields a unique action of Aut$(G)$ on the tree and thus canonicality.

The second main theorem (Theorem 2) extends the canonical construction to normal coverings. If $p\colon\widehat G\to G$ is a normal covering of a locally finite, connected graph $G$ and $\widehat G$ is chordal, then $\widehat G$ possesses a tree‑decomposition into its maximal cliques that is canonical with respect to the deck‑transformation group $\Gamma(p)$. This shows that the symmetry coming from the covering can be preserved while using maximal cliques as bags.

A significant application is given in Theorem 3, which characterises $r$‑locally chordal graphs (graphs whose balls of radius $r/2$ are chordal) via the unique canonical graph‑decomposition $\mathcal H_r(G)$ that displays the $r$‑global structure of $G$. Using the recent framework of $r$‑local coverings $G_r\to G$ (Diestel–Jacobs–Knappe–Kurkofka 2022), the authors note that $G$ is $r$‑locally chordal iff the $r$‑local covering $G_r$ is chordal. By Theorem 1, the canonical tree‑decomposition $T(G_r)$ of $G_r$ consists of cliques, and folding this decomposition back along the covering map yields $\mathcal H_r(G)$. Hence $\mathcal H_r(G)$ is a clique‑decomposition exactly when $G$ is $r$‑locally chordal.

The paper also delineates the limits of canonical decompositions. Example 5.1 shows that even finite connected chordal graphs may fail to admit a canonical decomposition into maximal cliques. Example 5.2 presents a countable chordal graph that has a clique‑decomposition but no canonical one, illustrating difficulties arising from infinite symmetry groups. Example 5.3 gives an uncountable chordal graph that admits a clique‑decomposition but cannot be decomposed into maximal cliques at all. These counterexamples demonstrate that canonicality cannot be universally strengthened to maximal‑clique decompositions, nor guaranteed for all infinite chordal graphs.

The paper concludes with a discussion of future directions: extending canonical decompositions to broader graph classes, exploring $r$‑local parameters such as tree‑width or tree‑depth, and investigating canonical decompositions for more general (possibly non‑normal) coverings. Overall, the work provides a robust bridge between chordal graph structure, tree‑decompositions, and graph symmetries, offering new tools for both local and global graph analysis.