Sphere and dot product representations of graphs

• Authors: Ross J. Kang;Tobias Müller
• Venue: Proceedings of the twenty-seventh annual symposium on Computational geometry
• Year: 2011

A graph G is a k-sphere graph if there are k-dimensional real vectors v1,..., vn such that ij ∈ E(G) if and only if the distance between vi and vj is at most 1. A graph G is a k-dot product graph if there are k-dimensional real vectors v1,...,vn such that ij ∈ E(G) if and only if the dot product of vi and vj is at least 1. By relating these two geometric graph constructions to oriented k-hyperplane arrangements, we prove that the problems of deciding, given a graph G, whether G is a k-sphere or a k-dot product graph are NP-hard for all k1. In the former case, this proves a conjecture of Breu and Kirkpatrick (1998). In the latter, this answers a question of Fiduccia, Scheinerman, Trenk and Zito (1998). Furthermore, motivated by the question of whether these two recognition problems are in NP, as well as by the Implicit Graph Conjecture, we demonstrate that, for all k 1, there exist k-sphere graphs and k-dot product graphs such that each representation in k-dimensional real vectors needs at least an exponential number of bits to be stored in the memory of a computer. On the other hand, we show that exponentially many bits are always enough. This resolves a question of Spinrad (2003).