On the computational complexity of degenerate unit distance representations of graphs

Details On the computational complexity of degenerate unit distance representations of graphs On the computational complexity of degenerate unit distance representations of graphs

2010-01-06

arxiv.org/abs/1001.0886v1

Mathematics

Combinatorics

11 pages, 6 figures

Scientific paper

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.

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