Refining the Hierarchies of Classes of Geometric Intersection Graphs

We analyse properties to show the strict containment between some natural classes of geometric intersection graphs. For example, we show that, for each integer k≥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, and the class of outer-segment graphs is a strict subclass of the class of outer-string graphs.