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Counterexamples to statements on isometric graph coverings

Paul Bastide, Julien Duron, Jędrzej Hodor, Weichan Liu, Xiangxiang Nie

TL;DR

The paper investigates whether structural bounds on a graph $G$ can be inferred when $G$ is edge-covered by a fixed number $k$ of connected isometric subgraphs. It provides negative results by constructing graphs $G_n$ with arbitrarily large treewidth that are edge-covered by $k$ isometric subgraphs for $k\in\{2,3,4\}$, using subdivided walls and incidence-coloring-based gadgets. The results rule out broad transfers of width or depth parameters under such coverings, even when the covering subgraphs are simple (e.g., radius-$2$ trees or basic split graphs), and highlight limits related to subdivision, degree constraints, and subcubic/twin-width phenomena.

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.

Counterexamples to statements on isometric graph coverings

TL;DR

The paper investigates whether structural bounds on a graph can be inferred when is edge-covered by a fixed number of connected isometric subgraphs. It provides negative results by constructing graphs with arbitrarily large treewidth that are edge-covered by isometric subgraphs for , using subdivided walls and incidence-coloring-based gadgets. The results rule out broad transfers of width or depth parameters under such coverings, even when the covering subgraphs are simple (e.g., radius- trees or basic split graphs), and highlight limits related to subdivision, degree constraints, and subcubic/twin-width phenomena.

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 is edge covered by connected isometric subgraphs , which properties of can we infer from properties of ? For example, Dumas, Foucaud, Perez and Todinca (SIDMA, 2024) proved that when are paths, then the pathwidth of is bounded in terms of . Among others, we show that there are graphs of arbitrarily large treewidth that can be isometrically edge covered by four trees.

Paper Structure

This paper contains 6 sections, 5 theorems, 1 equation, 3 figures.

Table of Contents

  1. Introduction
  2. Detailed statements
  3. Covering by trees
  4. Covering by three simple graphs
  5. Covering by two simple graphs
  6. Acknowledgments

Key Result

Theorem 1

For every positive integer $n$ and for every $k \in \{2,3,4\}$, there exist connected graphs $H_1,\dots,H_k$ such that and there exists a graph $G_n$ with $\mathop{\mathrm{\mathbf{tw}}}\nolimits(G_n) \geqslant n$ such that each of $H_1,\dots,H_k$ is an isometric subgraph of $G_n$ and $H_1,\ldots,H_k

Figures (3)

  • Figure 1: A wall $X$ of order $4$ along with a proper function mapping $\mathop{\mathrm{Inc}}\nolimits(X)$ to three colors: blue, pink, and yellow.
  • Figure 2: An illustration of the construction in the proof of \ref{['lem:proper-to-result']}. Vertices that are results of subdividing edges of the original wall are marked as squares. Vertices belonging to exactly one subgraph $H_i$ are color appropriately. Note that we used the function $\varphi$ as in \ref{['fig:wall-proper']}. Vertices $A = \{a_1,a_2,a_3,a_4\}$ are on the top of the figure. For clarity, we do not draw edges incident to vertices in $A$, instead, we only draw their beginnings and ends.
  • Figure 3: We use the same conventions as in \ref{['fig:wall']}. Note that the graphs $H_i$ corresponding to colors blue and yellow are trees of radius $2$, and the graph corresponding to color pink is in $\mathbb{A}(P_5)$.

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 5
  • proof
  • proof
  • proof : Proof of \ref{['thm:main-technical-any']}