# Refining the Hierarchies of Classes of Geometric Intersection Graphs

Keywords:
Geometric intersection graphs, Segment graphs, String graphs

### 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.