Suppose that A is a convex body in the plane and that A1, . . . , An are translates of A. Such translates give rise to an intersection graph of A, G = (V, E), with vertices V = {1, . . . , n} and edges E = {uv | Au ∩ Av = ∅}. The subgraph G = (V, E ) satisfying that E ⊂ E is the set of edges uv for which the interiors of Au and Av are disjoint is a unit distance graph of A. If furthermore G = G, i.e., if the interiors of Au and Av are disjoint whenever u = 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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.