Canonical tree-decompositions of chordal graphs

Raphael W. Jacobs; Paul Knappe

Canonical tree-decompositions of chordal graphs

Combinatorics 2026-05-05 v3

Authors: Raphael W. Jacobs , Paul Knappe

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.

Keywords

chordal graphs tree-decomposition canonical decomposition locally finite graphs clique decomposition graph automorphisms

Cite

@article{arxiv.2512.18480,
  title  = {Canonical tree-decompositions of chordal graphs},
  author = {Raphael W. Jacobs and Paul Knappe},
  journal= {arXiv preprint arXiv:2512.18480},
  year   = {2026}
}

Comments

v3: updated introduction; 20 pages, 1 figure; the results in this paper appeared in arXiv:2501.17320v1 which is now split into four separate papers

Canonical tree-decompositions of chordal graphs

Abstract

Keywords

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