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Abstract: Some graphs admit drawings in the Euclidean k-space in such a natu- ralway, that edges are represented as line segments of unit length. Such drawingswill be called k dimensional unit distance representations. When twonon-adjacent vertices are drawn in the same point, we say that therepresentation is degenerate. The dimension the Euclidean dimension of agraph is defined to be the minimum integer k needed that a given graph hasnon-degenerate k dimensional unit distance representation with the propertythat non-adjacent vertices are mapped to points, that are not distance oneappart. It is proved that deciding if an input graph is homomorphic to a graphwith dimension k >= 2 with the Euclidean dimension k >= 2 are NP-hardproblems.



Author: Jan Kratochvil, Boris Horvat, Tomaz Pisanski

Source: https://arxiv.org/







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