On the Computational Complexity of Degenerate Unit Distance Representations of Graphs

  • Boris Horvat
  • Jan Kratochvíl
  • Tomaž Pisanski
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6460)


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 non-adjacent 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.


unit distance graph the dimension of a graph the Euclidean dimension of a graph degenerate representation complexity 

Mathematics Subject Classification

05C62 05C12 68Q17 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Boris Horvat
    • 1
  • Jan Kratochvíl
    • 2
  • Tomaž Pisanski
    • 1
    • 3
  1. 1.IMFMUniversity of LjubljanaSlovenia
  2. 2.Charles UniversityPragueCzech Republic
  3. 3.IMFMUniversity of PrimorskaSlovenia

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