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Locally interval graphs are circular-arc graphs

Tara Abrishami, Sandra Albrechtsen, Nathan Bowler, Paul Knappe, Jana Katharina Nickel

TL;DR

The paper proves that for all integers $r\ge4$, a finite connected graph is $r$-locally interval if and only if it is an $r$-acyclic circular-arc graph, thereby linking local interval locality to global circular-arc representations. The authors leverage the theory of $r$-local coverings and a forbidden-subgraph framework, building on prior work on locally chordal graphs, to establish a 3-way equivalence among locality, global representations, and structural forbiddances. They extend known forbidden-subgraph characterizations to the $r$-locally interval setting and provide a precise treatment of the 3-locally chordal case, including a description via apex-augmented subgraphs. This work advances the local-global perspective in graph structure theory by connecting locality conditions to well-studied representation classes and by offering concrete characterization tools for both $r\ge4$ and $r=3$.

Abstract

Circular-arc graphs are graphs that can be represented as intersection graphs of subpaths of a cycle. Interval graphs are graphs that can be represented as intersection graphs of subpaths of a path. Since cycles are locally paths, every circular-arc graph is locally interval. In this paper, we prove that the converse holds as well: every locally interval graph is a circular-arc graph. This result and its proofs are connected to a recent broader study of structural local-global theory and build on previous work on locally chordal graphs.

Locally interval graphs are circular-arc graphs

TL;DR

The paper proves that for all integers , a finite connected graph is -locally interval if and only if it is an -acyclic circular-arc graph, thereby linking local interval locality to global circular-arc representations. The authors leverage the theory of -local coverings and a forbidden-subgraph framework, building on prior work on locally chordal graphs, to establish a 3-way equivalence among locality, global representations, and structural forbiddances. They extend known forbidden-subgraph characterizations to the -locally interval setting and provide a precise treatment of the 3-locally chordal case, including a description via apex-augmented subgraphs. This work advances the local-global perspective in graph structure theory by connecting locality conditions to well-studied representation classes and by offering concrete characterization tools for both and .

Abstract

Circular-arc graphs are graphs that can be represented as intersection graphs of subpaths of a cycle. Interval graphs are graphs that can be represented as intersection graphs of subpaths of a path. Since cycles are locally paths, every circular-arc graph is locally interval. In this paper, we prove that the converse holds as well: every locally interval graph is a circular-arc graph. This result and its proofs are connected to a recent broader study of structural local-global theory and build on previous work on locally chordal graphs.

Paper Structure

This paper contains 8 sections, 9 theorems, 3 figures.

Table of Contents

  1. Introduction
  2. Connections to local-global theory
  3. Background
  4. Locally chordal graphs
  5. Interval graphs
  6. The local covering
  7. Proof of Theorem 1
  8. Characterization of 3-locally chordal graphs

Key Result

Theorem 1.2

Let $G$ be a finite connected graph and $r \geqslant 4$ an integer. Then, $G$ is $r$-locally interval if and only if $G$ is $r$-acyclic circular-arc.

Figures (3)

  • Figure 1: Chordal minimal forbidden induced subgraphs of interval graphs. The dotted lines in \ref{['fig:dag']} and \ref{['fig:ddag']} represent paths of arbitrary length, except one of the dotted lines in \ref{['fig:dag']} must be of length at least one.
  • Figure 2: Non-chordal minimal forbidden induced subgraphs of $3$-acyclic circularc arc graphs, $n \geqslant 4$.
  • Figure 3: Additional forbidden induced subgraphs for $r$-acyclic circular-arc graphs in the case $r = 4$.

Theorems & Definitions (19)

  • Conjecture 1.1: Rzążewski 2024
  • Theorem 1.2
  • Theorem 1.3: LocallyChordal, Theorem 1 and localGlobalChordal, Theorem 1
  • Theorem 1
  • Lemma 2.1: localGlobalChordal, Lemma 3.6
  • Theorem 2.2: localGlobalChordal, Theorem 5
  • Theorem 2.3: Boland & Lekkeikerker ForbIndSubInterval, II
  • Lemma 2.4
  • proof
  • Theorem 3.1: Cao, Grippo & Safe ForbIndSubNHCAG, Theorem 1
  • ...and 9 more