On Comparable Box Dimension
Zdenek Dvor\'ak, Daniel Goncalves, Abhiruk Lahiri, Jane Tan, Torsten, Ueckerdt

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