The Three Dimensional Logic Engine

  • Matthew Kitching
  • Sue Whitesides
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3383)

Abstract

We consider the following graph embedding question: given a graph G, is it possible to map its vertices to points in 3D such that G is isomorphic to the mutual nearest neighbor graph of the set P of points to which the vertices are mapped? We show that this problem is NP-hard. We do this by extending the “logic engine” method to three dimensions by using building blocks inpired by the structure of diamond and by constructions of A.G. Bell and B. Fuller.

References

  1. 1.
    Bell, A.G.: Tetrahedral principle in kite structure. National Geographic Magazine 14(6), 219–251 (1903)Google Scholar
  2. 2.
    Buckminster Fuller, R.: Inventions, the Patented Works of R. Buckminster Fuller. St. Martin’s Press (1983)Google Scholar
  3. 3.
    Bose, P., Lenhart, W., Liotta, G.: Characterizing proximity trees. Algorithmica 16, 83–110 (1996)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Brandenburg, F.J., Eppstein, D., Goodrich, M.T., Kobourov, S.G., Liotta, G., Mutzel, P.: Selected open problems in graph drawing. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 515–539. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  5. 5.
    Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing,  ch. 11.2. Prentice Hall, Englewood Cliffs (1999)Google Scholar
  6. 6.
    Dillencourt, M.B.: Realizability of Delaunay triangulations. Informa. Process. Lett. 33(6), 283–287 (1990)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dillencourt, M.B., Smith, W.D.: Graph-theoretical conditions for inscribability and Delaunay realizability. In: Proc. 6th Canad. Conf. Comput. Geom., pp. 287–292 (1994)Google Scholar
  8. 8.
    Eades, P., Whitesides, S.: The logic engine and the realization problem for nearest neighbour graphs. Theoretical Computer Science 169, 23–37 (1996)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Eades, P., Whitesides, S.: The realization problem for Euclidean minimum spanning trees is NP-hard. Algorithmica 16, 60–82 (1996)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Garey, M., Johnson, D.: Computers and Intractability: a Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)MATHGoogle Scholar
  11. 11.
    Jaromczyk, J.W., Toussaint, G.T.: Relative neighborhood graphs and their relatives. Proc. IEEE 80(9), 1502–1517 (1992)CrossRefGoogle Scholar
  12. 12.
    Lenhart, W., Liotta, G.: The drawability problem for minimum weight triangulations. Theoret. Comp. Sci. 27, 261–286 (2002)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Liotta, G., Di Battista, G.: Computing proximity drawings of trees in the 3-dimensional space. In: Sack, J.-R., Akl, S.G., Dehne, F., Santoro, N. (eds.) WADS 1995. LNCS, vol. 955, pp. 239–250. Springer, Heidelberg (1995)Google Scholar
  14. 14.
    Liotta, G., Lubiw, A., Meijer, H., Whitesides, S.H.: The rectangle of influence drawability problem. Comput. Geom. Theory Appl. 10(1), 1–22 (1998)MATHMathSciNetGoogle Scholar
  15. 15.
    Liotta, G., Meijer, H.: Drawing of trees. Computational Geometry: Theory and Applications 24(3), 147–178 (2003)MATHMathSciNetGoogle Scholar
  16. 16.
    Toussaint, G.: A graph-theoretical primal sketch. In: Computational Morphology, pp. 229–260. North-Holland, Amsterdam (1988)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Matthew Kitching
    • 1
  • Sue Whitesides
    • 2
  1. 1.Dept. of Comp. Sci.U. of TorontoCanada
  2. 2.School of Comp. Sci.McGill U.MontrealCanada

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