Refining the Hierarchies of Classes of Geometric Intersection Graphs

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.