Distinguishing classes of intersection graphs of homothets or similarities of two convex disks

TL;DR

This paper classifies intersection graphs of homothets and similarities of smooth convex disks, showing they are uniquely determined by affine or similarity equivalence of the disks.

Contribution

It provides a complete classification of intersection graph classes for homothets and similarities of convex disks based on affine and similarity equivalences.

Findings

Intersection graphs of homothets are classified by affine equivalence.

Intersection graphs of similarities are classified by similarity.

The classification is complete for smooth convex disks.

Abstract

For smooth convex disks $A$, i.e., convex compact subsets of the plane with non-empty interior, we classify the classes $G^{\text{hom}}(A)$ and $G^{\text{sim}}(A)$ of intersection graphs that can be obtained from homothets and similarities of $A$, respectively. Namely, we prove that $G^{\text{hom}}(A)=G^{\text{hom}}(B)$ if and only if $A$ and $B$ are affine equivalent, and $G^{\text{sim}}(A)=G^{\text{sim}}(B)$ if and only if $A$ and $B$ are similar.