On the computational complexity of degenerate unit distance representations of graphs

Authors:
Boris Horvat;Jan Kratochvíl;Tomaž Pisanski
Affiliations:
IMFM, University of Ljubljana, Slovenia,;Charles University, Prague, Czech Republic;IMFM, University of Ljubljana and University of Primorska, Slovenia
Venue:
IWOCA'10 Proceedings of the 21st international conference on Combinatorial algorithms
Year:
2010

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Abstract

Some graphs admit drawings in the Euclidean k-space in such a (natural) way, that edges are represented as line segments of unit length. Such embeddings are called k-dimensional unit distance representations. The embedding is strict if the distances of points representing nonadjacent pairs of vertices are different than 1.When two nonadjacent vertices are drawn in the same point, we say that the representation is degenerate. Computational complexity of nondegenerate embeddings has been studied before. We initiate the study of the computational complexity of (possibly) degenerate embeddings. In particular we prove that for every k≥ 2, deciding if an input graph has a (possibly) degenerate k-dimensional unit distance representation is NP-hard.