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arXiv · 2511.03524

Counterexamples to statements on isometric graph coverings

Paul Bastide·Julien Duron·Jędrzej Hodor·Weichan Liu·Xiangxiang Nie
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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.

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Paul Bastide, Julien Duron, Jędrzej Hodor, Weichan Liu, Xiangxiang Nie. 2025-11-05. Counterexamples to statements on isometric graph coverings. https://arxiv.org/abs/2511.03524

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