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.