Refining the Hierarchies of Classes of Geometric Intersection Graphs

TL;DR

This paper investigates the hierarchy of geometric intersection graph classes, establishing strict containments and characterizations among classes such as outerplanar, outer-segment, and outer-string graphs, based on segment lengths and disk sizes.

Contribution

It provides new insights into the strict hierarchy and containment relations among various classes of geometric intersection graphs, including segments and disks with varying parameters.

Findings

Outerplanar graphs characterized via 1-subdivision and outer-segment graphs.

Strict hierarchy of segment intersection graphs with increasing segment lengths.

Strict hierarchy of disk intersection graphs with increasing disk sizes.

Abstract

We analyse properties of geometric intersection graphs to show the strict containment between some natural classes of geometric intersection graphs. In particular, we show the following properties: - A graph $G$ is outerplanar if and only if the 1-subdivision of $G$ is outer-segment. - For each integer $k\ge 1$, the class of intersection graphs of segments with $k$ different lengths is a strict subclass of the class of intersection graphs of segments with $k+1$ different lengths. - For each integer $k\ge 1$, the class of intersection graphs of disks with $k$ different sizes is a strict subclass of the class of intersection graphs of disks with $k+1$ different sizes. - The class of outer-segment graphs is a strict subclass of the class of outer-string graphs.