Intersection Graphs with and without Product Structure
Laura Merker, Lena Scherzer, Samuel Schneider, and Torsten Ueckerdt

TL;DR
This paper investigates when classes of intersection graphs of geometric objects admit product structure, identifying a threshold parameter that determines the presence or absence of such structure for various shapes.
Contribution
It establishes necessary and sufficient conditions for intersection graph classes to admit product structure based on a parameter lpha, and characterizes this for multiple geometric sets.
Findings
For many shapes, lpha^*(S) = 1, so no product structure except contact graphs.
For some shapes, 0 < lpha^*(S) < 1, indicating a threshold behavior.
Identifies geometric conditions influencing the existence of product structure in intersection graphs.
Abstract
A graph class admits product structure if there exists a constant such that every is a subgraph of for a path and some graph of treewidth . Famously, the class of planar graphs, as well as many beyond-planar graph classes are known to admit product structure. However, we have only few tools to prove the absence of product structure, and hence know of only a few interesting examples of classes. Motivated by the transition between product structure and no product structure, we investigate subclasses of intersection graphs in the plane (e.g., disk intersection graphs) and present necessary and sufficient conditions for these to admit product structure. Specifically, for a set (e.g., a disk) and a real number , we consider intersection graphs of -free homothetic copies of .…
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