Grid Intersection Graphs and Order Dimension

TL;DR

This paper explores the properties of grid intersection graphs through the lens of order dimension, establishing bounds and class relations that deepen understanding of their structure.

Contribution

It introduces bounds on the order dimension of certain grid intersection graphs and studies the hierarchy of related graph classes.

Findings

Partial orders of height two with grid intersection comparability graphs have dimension ≤4.

Provides a detailed analysis of classes between grid intersection graphs and bipartite permutation graphs.

Highlights the role of order dimension in understanding graph class containment and properties.

Abstract

We study subclasses of grid intersection graphs from the perspective of order dimension. We show that partial orders of height two whose comparability graph is a grid intersection graph have order dimension at most four. Starting from this observation we provide a comprehensive study of classes of graphs between grid intersection graphs and bipartite permutation graphs and the containment relation on these classes. Order dimension plays a role in many arguments.