On tree-decompositions for infinite chordal graphs

TL;DR

This paper extends the theory of tree-decompositions for infinite chordal graphs, providing new characterizations and conditions for decompositions into maximal and finite cliques, using greedy constructions and minimal separator theorems.

Contribution

It introduces two main results: one strengthening conditions for tree-decompositions into maximal cliques, and another characterizing graphs with finite clique decompositions, including forbidden minor descriptions.

Findings

Chordal graphs without a strict comb admit tree-decompositions into maximal cliques.

Connected chordal graphs with finite clique decompositions are characterized by forbidden minors.

The proofs utilize greedy constructions and Halin's minimal separator theorem.

Abstract

A graph is chordal if it contains no induced cycle of length four or more. While finite chordal graphs are precisely those admitting tree-decompositions into cliques, this fails for infinite graphs. We establish two results extending the known theory to the infinite setting. Our first result strengthens sufficient conditions of Halin, K\v{r}\'i\v{z}-Thomas, and Chudnovsky-Nguyen-Scott-Seymour: We show that every chordal graph without a strict comb of cliques admits a tree-decomposition into maximal cliques. Our second result characterises the chordal graphs admitting tree-decompositions into finite cliques: a connected graph admits such a decomposition if and only if it is chordal, admits a normal spanning tree, and does not contain $\mathcal{H}$ $\unicode{x2013}$ an infinite clique with two non-adjacent dominating vertices $\unicode{x2013}$ as an induced minor. Combined with the characterisation of graphs with normal spanning trees, this yields a description by three types of forbidden minors. Both proofs proceed via greedy constructions of length $\omega$, with the key new ingredient for the second result being an Extension Lemma that uses a finiteness theorem of Halin on minimal separators to produce suitable finite clique extensions at each step.