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The global structure of locally chordal graphs

Tara Abrishami, Paul Knappe

TL;DR

<3-5 sentence high-level summary> The paper investigates how the local structure of r-locally chordal graphs determines their global organization. It develops two complementary frameworks—region representations (r-acyclic region intersection graphs) and graph-decompositions into cliques—showing they are equivalent descriptions of the same global structure and that this structure witnesses local chordality around every vertex. A central achievement is proving that for every r≥3, a graph is r-locally chordal if and only if it admits an r-acyclic region representation (and an r-acyclic decomposition into cliques), with efficient algorithms to compute these representations via r-local covers and foldings. These results generalize chordal graph theory to locally chordal graphs and establish a foundation for new width parameters and algorithmic applications, including parallel computation of clique graphs and potential MIS algorithms on locally chordal graphs.

Abstract

A graph is locally chordal if each of its small-radius balls is chordal. In an earlier work [AKK25], the authors and Kobler proved that locally chordal graphs can be characterized by having chordal local covers, by forbidding short cycles and wheels as induced subgraphs, and by the property that each of their minimal local separators is a clique. In this paper, we address the global structure of locally chordal graphs. The global structure of chordal graphs is given by the following characterizations: a graph is chordal if and only if it is the intersection graph of subtrees of a tree, if and only if it admits a tree-decomposition into cliques. We prove a local analog of this characterization, which essentially says that a graph is locally chordal if and only if it is the intersection graph of special subtrees of a high-girth graph, if and only if it admits a special graph-decomposition over a high-girth graph into cliques. We also prove that these global representations of locally chordal graphs can be efficiently computed. This paper has two major contributions. The first is to exhibit for locally chordal graphs an ideal "local to global" analysis: given a graph class defined by restricted local structure, we fully describe the global structure of graphs in the class. The second is to develop the theory of graph-decompositions. Much of the work in this paper is devoted to properties of graph-decompositions that represent the global structure of graphs. This theory will be useful to find global decompositions for graph classes beyond locally chordal graphs.

The global structure of locally chordal graphs

TL;DR

<3-5 sentence high-level summary> The paper investigates how the local structure of r-locally chordal graphs determines their global organization. It develops two complementary frameworks—region representations (r-acyclic region intersection graphs) and graph-decompositions into cliques—showing they are equivalent descriptions of the same global structure and that this structure witnesses local chordality around every vertex. A central achievement is proving that for every r≥3, a graph is r-locally chordal if and only if it admits an r-acyclic region representation (and an r-acyclic decomposition into cliques), with efficient algorithms to compute these representations via r-local covers and foldings. These results generalize chordal graph theory to locally chordal graphs and establish a foundation for new width parameters and algorithmic applications, including parallel computation of clique graphs and potential MIS algorithms on locally chordal graphs.

Abstract

A graph is locally chordal if each of its small-radius balls is chordal. In an earlier work [AKK25], the authors and Kobler proved that locally chordal graphs can be characterized by having chordal local covers, by forbidding short cycles and wheels as induced subgraphs, and by the property that each of their minimal local separators is a clique. In this paper, we address the global structure of locally chordal graphs. The global structure of chordal graphs is given by the following characterizations: a graph is chordal if and only if it is the intersection graph of subtrees of a tree, if and only if it admits a tree-decomposition into cliques. We prove a local analog of this characterization, which essentially says that a graph is locally chordal if and only if it is the intersection graph of special subtrees of a high-girth graph, if and only if it admits a special graph-decomposition over a high-girth graph into cliques. We also prove that these global representations of locally chordal graphs can be efficiently computed. This paper has two major contributions. The first is to exhibit for locally chordal graphs an ideal "local to global" analysis: given a graph class defined by restricted local structure, we fully describe the global structure of graphs in the class. The second is to develop the theory of graph-decompositions. Much of the work in this paper is devoted to properties of graph-decompositions that represent the global structure of graphs. This theory will be useful to find global decompositions for graph classes beyond locally chordal graphs.

Paper Structure

This paper contains 42 sections, 62 theorems, 5 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Characterization via region representations
  3. Characterization via graph-decompositions
  4. Locally inducing tree-decompositions
  5. Inducing local separations
  6. Further results: Computing region representations
  7. Acknowledgement
  8. Notation and conventions
  9. Equivalence: Graph-decompositions and region representations
  10. Graph-decompositions and region intersection graphs
  11. Canonical graph-decompositions & region representations
  12. Clique graphs and graph-decompositions into maximal cliques
  13. Canonical clique graphs
  14. Overview: Region representations of (locally) chordal graphs
  15. Proof of Theorem XX
  16. ...and 27 more sections

Key Result

Theorem 1

Let $G$ be a locally finiteRecall that a graph is locally finite if each vertex has finite degree. In particular, finite graphs are locally finite. graph and $r \geqslant 3$ an integer. The following are equivalent:

Figures (2)

  • Figure 1: A locally chordal graph $G$, depicted in black, obtained by gluing cliques in the shape of $H$, depicted in blue.
  • Figure 2: Examples of local separations $\{E_0,X,E_1\}$. The partition classes of $\{E_0,E_1\}$ are depicted in red and blue, respectively. The set $X$ is formed by the yellow vertices.

Theorems & Definitions (116)

  • Theorem 1
  • Theorem 1.1: LocallyChordal, Theorem 1
  • Lemma 1.2: LocallyChordal, Lemma 2.4
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2
  • Lemma 2.3
  • Lemma 2.4
  • Proposition 3.1: Folklore
  • Theorem 3.2
  • ...and 106 more