On the computational complexity of degenerate unit distance representations of graphs
Abstract
Some graphs admit drawings in the Euclidean k-space in such a (natu- ral) way, that edges are represented as line segments of unit length. Such drawings will be called k dimensional unit distance representations. When two non-adjacent vertices are drawn in the same point, we say that the representation is degenerate. The dimension (the Euclidean dimension) of a graph is defined to be the minimum integer k needed that a given graph has non-degenerate k dimensional unit distance representation (with the property that non-adjacent vertices are mapped to points, that are not distance one appart). It is proved that deciding if an input graph is homomorphic to a graph with dimension k >= 2 (with the Euclidean dimension k >= 2) are NP-hard problems.
Keywords
Cite
@article{arxiv.1001.0886,
title = {On the computational complexity of degenerate unit distance representations of graphs},
author = {Jan Kratochvil and Boris Horvat and Tomaz Pisanski},
journal= {arXiv preprint arXiv:1001.0886},
year = {2010}
}
Comments
11 pages, 6 figures