Advertisement

Visibility Representations of Boxes in 2.5 Dimensions

  • Alessio Arleo
  • Carla Binucci
  • Emilio Di Giacomo
  • William S. Evans
  • Luca Grilli
  • Giuseppe Liotta
  • Henk Meijer
  • Fabrizio Montecchiani
  • Sue Whitesides
  • Stephen Wismath
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9801)

Abstract

We initiate the study of 2.5D box visibility representations (2.5D-BR) where vertices are mapped to 3D boxes having the bottom face in the plane \(z=0\) and edges are unobstructed lines of sight parallel to the x- or y-axis. We prove that: (i) Every complete bipartite graph admits a 2.5D-BR; (ii) The complete graph \(K_n\) admits a 2.5D-BR if and only if \(n \leqslant 19\); (iii) Every graph with pathwidth at most 7 admits a 2.5D-BR, which can be computed in linear time. We then turn our attention to 2.5D grid box representations (2.5D-GBR) which are 2.5D-BRs such that the bottom face of every box is a unit square at integer coordinates. We show that an n-vertex graph that admits a 2.5D-GBR has at most \(4n - 6 \sqrt{n}\) edges and this bound is tight. Finally, we prove that deciding whether a given graph G admits a 2.5D-GBR with a given footprint is NP-complete. The footprint of a 2.5D-BR \(\varGamma \) is the set of bottom faces of the boxes in \(\varGamma \).

Keywords

Visibility Representation Complete Graph Geometric Object Hamiltonian Path Complete Bipartite Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Ahmed, M.E., Yusuf, A.B., Polin, M.Z.H.: Bar 1-visibility representation of optimal 1-planar graph. Elect. Inf. Comm. Technol. (EICT) 2013, 1–5 (2014)Google Scholar
  2. 2.
    Akiyama, T., Nishizeki, T., Saito, N.: NP-completeness of the hamiltonian cycle problem for bipartite graphs. J. Inf. Process. 3(2), 73–76 (1980)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Arleo, A., Binucci, C., Di Giacomo, E, Evans, W.S., Grilli, L., Liotta, G., Meijer, H., Montecchiani, F., Whitesides, S., Wismath, S.: Visibility representations of boxes in 2.5 dimensions. CoRR, abs/1608.08899 (2016)Google Scholar
  4. 4.
    Biedl, T., Liotta, G., Montecchiani, F.: On visibility representations of non-planar graphs. In: Fekete, S., Lubiw, A., (eds.) SoCG 2016, vol. LIPICs, pp. 19:1–19:16. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2016)Google Scholar
  5. 5.
    Bose, P., Dean, A., Hutchinson, J., Shermer, T.: On rectangle visibility graphs. In: North, S. (ed.) GD 1996. LNCS, vol. 1190, pp. 25–44. Springer, Heidelberg (1997). doi: 10.1007/3-540-62495-3_35 CrossRefGoogle Scholar
  6. 6.
    Bose, P., Everett, H., Fekete, S.P., Houle, M.E., Lubiw, A., Meijer, H., Romanik, K., Rote, G., Shermer, T.C., Whitesides, S., Zelle, C.: A visibility representation for graphs in three dimensions. J. Graph Algorithms Appl. 2(3), 1–16 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brandenburg, F.: 1-visibility representations of 1-planar graphs. J. Graph Algorithms Appl. 18(3), 421–438 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bruckdorfer, T., Kaufmann, M., Montecchiani, F.: 1-bend orthogonal partial edge drawing. J. Graph Algorithms Appl. 18(1), 111–131 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Carmi, P., Friedman, E., Katz, M.J.: Spiderman graph: visibility in urban regions. Comput. Geometry 48(3), 251–259 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dean, A.M., Ellis-Monaghan, J.A., Hamilton, S., Pangborn, G.: Unit rectangle visibility graphs. Electr. J. Comb. 15(1), 1–24 (2008)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Dean, A.M., Evans, W., Gethner, E., Laison, J.D., Safari, M.A., Trotter, W.T.: Bar \(k\)-visibility graphs. J. Graph Algorithms Appl. 11(1), 45–59 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dean, A.M., Hutchinson, J.P.: Rectangle-visibility representations of bipartite graphs. Discrete Appl. Math. 75(1), 9–25 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dean, A.M., Hutchinson, J.P.: Rectangle-visibility layouts of unions and products of trees. J. Graph Algorithms Appl. 2(8), 1–21 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dean, A.M., Veytsel, N.: Unit bar-visibility graphs. Congr. Num. 160, 161–175 (2003)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Di Giacomo, E., Didimo, W., Evans, W.S., Liotta, G., Meijer, H., Montecchiani, F., Wismath, S.K.: Ortho-polygon visibility representations of embedded graphs. In: Nöllenburg, M., Hu, Y. (eds.) GD 2016. LNCS, vol. 9801, pp. 280–294. Springer, Heidelberg (2016)Google Scholar
  16. 16.
    Duchet, P., Hamidoune, Y., Las, M., Vergnas, H.M.: Representing a planar graph by vertical lines joining different levels. Discrete Math. 46(3), 319–321 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Evans, W., Kaufmann, M., Lenhart, W., Mchedlidze, T., Wismath, S.: Bar 1-visibility graphs and their relation to other nearly planar graphs. J. Graph Algorithms Appl. 18(5), 721–739 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Cobos, F.J., Dana, J.C., Hurtado, F., Márquez, A., Mateos, F.: On a visibility representation of graphs. In: Brandenburg, F.J. (ed.) GD 1995. LNCS, vol. 1027, pp. 152–161. Springer, Heidelberg (1996). doi: 10.1007/BFb0021799 CrossRefGoogle Scholar
  19. 19.
    Fekete, S.P., Meijer, H.: Rectangle and box visibility graphs in 3D. Int. J. Comput. Geometry Appl. 9(1), 1–28 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Felsner, S., Massow, M.: Parameters of bar \(k\)-visibility graphs. J. Graph Algorithms Appl. 12(1), 5–27 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Garey, M.R., Johnson, D.S., So, H.C.: An application of graph coloring to printed circuit testing. IEEE Trans. Circuits Syst. CAS–23(10), 591–599 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gupta, A., Nishimura, N., Proskurowski, A., Ragde, P.: Embeddings of \(k\)-connected graphs of pathwidth \(k\). Discrete Appl. Math. 145(2), 242–265 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hutchinson, J.P., Shermer, T., Vince, A.: On representations of some thickness-two graphs. Comp. Geometry 13(3), 161–171 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kant, G., Liotta, G., Tamassia, R., Tollis, I.G.: Area requirement of visibility representations of trees. Inf. Process. Lett. 62(2), 81–88 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kleitman, J.D., Gyárfás, A., Tóth, G.: Convex sets in the plane with three of every four meeting. Combinatorica 21(2), 221–232 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Liotta, G., Montecchiani, F.: L-visibility drawings of IC-planar graphs. Inf. Process. Lett. 116(3), 217–222 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Otten, R.H.J.M., Van Wijk, J.G.: Graph representations in interactive layout design. In: IEEE ISCSS, pp. 91–918. IEEE (1978)Google Scholar
  28. 28.
    Rosenstiehl, P., Tarjan, R.E.: Rectilinear planar layouts and bipolar orientations of planar graphs. Discrete Comput. Geom. 1, 343–353 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Shermer, T.C.: On rectangle visibility graphs III. external visibility and complexity. In: Canadian Conference on Computational Geometry, pp. 234–239 (1996)Google Scholar
  30. 30.
    Štola, J.: Unimaximal sequences of pairs in rectangle visibility drawing. In: Tollis, I.G., Patrignani, M. (eds.) GD 2008. LNCS, vol. 5417, pp. 61–66. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-00219-9_7 CrossRefGoogle Scholar
  31. 31.
    Streinu, I., Whitesides, S.: Rectangle visibility graphs: characterization, construction, and compaction. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 26–37. Springer, Heidelberg (2003). doi: 10.1007/3-540-36494-3_4 CrossRefGoogle Scholar
  32. 32.
    Sultana, S., Rahman, M.S., Roy, A., Tairin, S.: Bar 1-visibility drawings of 1-planar graphs. In: Gupta, P., Zaroliagis, C. (eds.) ICAA 2014. LNCS, pp. 62–76. Springer International Publishing, New York (2014). doi: 10.1007/978-3-319-04126-1_6 Google Scholar
  33. 33.
    Tamassia, R., Tollis, I.G.: A unified approach to visibility representations of planar graphs. Discrete Comput. Geom. 1(1), 321–341 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Tamassia, R., Tollis, I.G.: Representations of graphs on a cylinder. SIAM J. Discrete Math. 4(1), 139–149 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Thomassen, C.: Plane representations of graphs. In: Progress in Graph Theory, pp. 43–69. AP (1984)Google Scholar
  36. 36.
    Wismath, S.K.: Characterizing bar line-of-sight graphs. In: Proceedings of 1st Symposium on Computational Geometry, pp. 147–152 (1985)Google Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Alessio Arleo
    • 1
  • Carla Binucci
    • 1
  • Emilio Di Giacomo
    • 1
  • William S. Evans
    • 2
  • Luca Grilli
    • 1
  • Giuseppe Liotta
    • 1
  • Henk Meijer
    • 3
  • Fabrizio Montecchiani
    • 1
  • Sue Whitesides
    • 4
  • Stephen Wismath
    • 5
  1. 1.Università degli Studi di PerugiaPerugiaItaly
  2. 2.University of British ColumbiaVancouverCanada
  3. 3.University College RooseveltMiddelburgThe Netherlands
  4. 4.University of VictoriaVictoriaCanada
  5. 5.University of LethbridgeLethbridgeCanada

Personalised recommendations