Bidimensionality of Geometric Intersection Graphs

TL;DR

This paper establishes a linear relationship between treewidth and grid minors in geometric intersection graphs, extending bidimensionality theory to a broad class of geometric graphs and enabling advanced algorithmic applications.

Contribution

It proves that geometric intersection graphs with certain restrictions have a linear relation between treewidth and grid minors, broadening the scope of bidimensionality theory.

Findings

Linear relation between treewidth and grid minors in geometric intersection graphs.

Extension of bidimensionality theory to non-convex geometric objects.

Applicability to a wide range of geometric graph classes.

Abstract

Let B be a finite collection of geometric (not necessarily convex) bodies in the plane. Clearly, this class of geometric objects naturally generalizes the class of disks, lines, ellipsoids, and even convex polygons. We consider geometric intersection graphs GB where each body of the collection B is represented by a vertex, and two vertices of GB are adjacent if the intersection of the corresponding bodies is non-empty. For such graph classes and under natural restrictions on their maximum degree or subgraph exclusion, we prove that the relation between their treewidth and the maximum size of a grid minor is linear. These combinatorial results vastly extend the applicability of all the meta-algorithmic results of the bidimensionality theory to geometrically defined graph classes.