On Comparable Box Dimension

TL;DR

This paper introduces the concept of comparable box dimension for graphs, establishing bounds for minor-closed classes and exploring its properties in geometric graph representations.

Contribution

It defines comparable box dimension for graphs and proves that proper minor-closed classes have bounded dimensions, advancing understanding of geometric graph representations.

Findings

Proper minor-closed classes have bounded comparable box dimensions

Introduces the notion of comparable box dimension for graphs

Explores properties and implications of this new dimension concept

Abstract

Two boxes in $\mathbb{R}^d$ are comparable if one of them is a subset of a translation of the other one. The comparable box dimension of a graph $G$ is the minimum integer $d$ such that $G$ can be represented as a touching graph of comparable axis-aligned boxes in $\mathbb{R}^d$. We show that proper minor-closed classes have bounded comparable box dimensions and explore further properties of this notion.