IWOCA 2010: Combinatorial Algorithms pp 274-285

# On the Computational Complexity of Degenerate Unit Distance Representations of Graphs

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

## 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 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.

## Keywords

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

## Mathematics Subject Classification

05C62 05C12 68Q17

## References

1. 1.
Boben, M., Pisanski, T.: Polycyclic configurations. European J. Combin. 24(4), 431–457 (2003)
2. 2.
Buckley, F., Harary, F.: On the Euclidean dimension of a Wheel. Graphs Combin. 4, 23–30 (1988)
3. 3.
Cabello, S., Demaine, E., Rote, G.: Planar embeddings of graphs with specified edge lengths. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 283–294. Springer, Heidelberg (2004)
4. 4.
Eades, P., Whitesides, S.: The logic engine and the realization problem for nearest neighbor graphs. Theor. Comput. Sc. 169(1), 23–37 (1996)
5. 5.
Eades, P., Wormald, N.C.: Fixed edge-length graph drawing is NP-hard. Discrete Appl. Math. 28(2), 111–134 (1990)
6. 6.
Eppstein, D.: Blog entry (January 2010), http://11011110.livejournal.com/188807.html
7. 7.
Erdös, P., Harary, F., Tutte, W.T.: On the dimension of a graph. Mathematika 12, 118–122 (1965)
8. 8.
Hendrickson, B.: Conditions For Unique Graph Realizations. SIAM J. Comput. 21(1), 65–84 (1992)
9. 9.
Horvat, B., Pisanski, T.: Unit distance representations of the Petersen graph in the plane. Ars Combin. (to appear)Google Scholar
10. 10.
Žitnik, A., Horvat, B., Pisanski, T.: All generalized Petersen graphs are unit-distance graphs (submitted)Google Scholar
11. 11.
Lovász, L.: Self-dual polytopes and the chromatic number of distance graphs on the sphere. Acta Sci. Math. (Szeged) 45(1-4), 317–323 (1983)
12. 12.
Maehara, H.: On the Euclidean dimension of a complete multipartite graph. Discrete Math. 72, 285–289 (1988)
13. 13.
Maehara, H.: Note on Induced Subgraphs of the Unit Distance Graph. Discrete Comput. Geom. 4, 15–18 (1989)
14. 14.
Maehara, H., Rödl, V.: On the Dimension to Represent a Graph by a Unit Distance Graph. Graphs Combin. 6, 365–367 (1990)
15. 15.
Renegar, J.: On the computational complexity and geometry of the first-order theory of the reals, part I: Introduction. Preliminaries. The geometry of semi-algebraic sets. The decision problem for the exitential theory of the reals. J. Symb. Comput. 13, 255–300 (1992)
16. 16.
Soifer, A.: The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators. Springer, Heidelberg (2008)

## 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