Counterexamples to statements on isometric graph coverings

TL;DR

This paper presents counterexamples to certain assumptions about how properties of a graph can be inferred from isometric coverings by subgraphs, challenging previous conjectures and demonstrating limitations of such inferences.

Contribution

It provides counterexamples to claims that properties like treewidth can be bounded based on isometric coverings by specific subgraphs, revealing new limitations in graph covering theory.

Findings

Graphs of arbitrarily large treewidth can be covered by four isometric trees.

Counterexamples show properties like bounded treewidth do not necessarily follow from isometric coverings.

Challenges to previous conjectures about property inheritance in isometric graph coverings.

Abstract

A connected subgraph of a graph is isometric if it preserves distances. In this short note, we provide counterexamples to several variants of the following general question: When a graph $G$ is edge covered by connected isometric subgraphs $H_1,\dots,H_k$, which properties of $G$ can we infer from properties of $H_1,\dots,H_k$? For example, Dumas, Foucaud, Perez and Todinca (SIDMA, 2024) proved that when $H_1,\dots,H_k$ are paths, then the pathwidth of $G$ is bounded in terms of $k$. Among others, we show that there are graphs of arbitrarily large treewidth that can be isometrically edge covered by four trees.