Classifying Convex Bodies by their Contact and Intersection Graphs

Suppose that $A$ is a convex body in the plane and that $A_1,\dots,A_n$ are translates of $A$. Such translates give rise to an intersection graph of $A$, $G=(V,E)$, with vertices $V=\{1,\dots,n\}$ and edges $E=\{uv\mid A_u\cap A_v\neq \emptyset\}$. The subgraph $G'=(V, E')$ satisfying that $E'\subset E$ is the set of edges $uv$ for which the interiors of $A_u$ and $A_v$ are disjoint is a unit distance graph of $A$. If furthermore $G'=G$, i.e., if the interiors of $A_u$ and $A_v$ are disjoint whenever $u\neq v$, then $G$ is a contact graph of $A$. In this paper we study which pairs of convex bodies have the same contact, unit distance, or intersection graphs. We say that two convex bodies $A$ and $B$ are equivalent if there exists a linear transformation $B'$ of $B$ such that for any slope, the longest line segments with that slope contained in $A$ and $B'$, respectively, are equally long. For a broad class of convex bodies, including all strictly convex bodies and linear transformations of regular polygons, we show that the contact graphs of $A$ and $B$ are the same if and only if $A$ and $B$ are equivalent. We prove the same statement for unit distance and intersection graphs.