11/5/2023 0 Comments Dot product 4d sphere![]() Graphs of sphericity one are also called unit interval graphs, and graphs of sphericity at most two are called unit disk graphs. thesis to study the geometry of molecules. (Every graph has finite sphericity .) Sphericity was first introduced by Havel in his Ph.D. The sphericity sph( G) of a graph G is the least k such that G is a k-sphere graph. Alternatively, a k-sphere graph can be seen as the intersection graph of equal radius spheres in k-dimensional space that is, we can represent each vertex i by a sphere S i⊆ℝ k of radius 1 in such a way that ij∈ E( G) if and only if S i∩ S j≠∅. This resolves a question of Spinrad (Efficient Graph Representations, 2003).Ī graph G is a k-sphere graph if there are vectors v 1,…, v n∈ℝ k such that ∥ v i− v j∥≤1 if and only if ij∈ E( G). On the other hand, we show that exponentially many bits are always enough. 181:113–138, 1998).įurthermore, 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. In the latter, this answers a question of Fiduccia et al. In the former case, this proves a conjecture of Breu and Kirkpatrick (Comput. A graph G is a k-dot product graph if there are k-dimensional real vectors v 1,…, v n such that ij∈ E( G) if and only if the dot product of v i and v j is at least 1.īy 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 k>1. A graph G is a k-sphere graph if there are k-dimensional real vectors v 1,…, v n such that ij∈ E( G) if and only if the distance between v i and v j is at most 1.
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