Grid-Obstacle Representations with Connections to Staircase Guarding

  • Therese BiedlEmail author
  • Saeed Mehrabi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10692)


In this paper, we study grid-obstacle representations of graphs where we assign grid-points to vertices and define obstacles such that an edge exists if and only if an xy-monotone grid path connects the two endpoints without hitting an obstacle or another vertex. It was previously argued that all planar graphs have a grid-obstacle representation in 2D, and all graphs have a grid-obstacle representation in 3D. In this paper, we show that such constructions are possible with significantly smaller grid-size than previously achieved. Then we study the variant where vertices are not blocking, and show that then grid-obstacle representations exist for bipartite graphs. The latter has applications in so-called staircase guarding of orthogonal polygons; using our grid-obstacle representations, we show that staircase guarding is NP-hard in 2D.


  1. 1.
    Alimonti, P., Kann, V.: Some APX-completeness results for cubic graphs. Theor. Comput. Sci. 237(1–2), 123–134 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alpert, H., Koch, C., Laison, J.D.: Obstacle numbers of graphs. Discrete Comput. Geom. 44(1), 223–244 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Biedl, T.: Small drawings of outerplanar graphs, series-parallel graphs, and other planar graphs. Discrete Comput. Geom. 45(1), 141–160 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Biedl, T.: Height-preserving transformations of planar graph drawings. In: Duncan, C., Symvonis, A. (eds.) GD 2014. LNCS, vol. 8871, pp. 380–391. Springer, Heidelberg (2014). Google Scholar
  5. 5.
    Biedl, T., Kaufmann, M., Mutzel, P.: Drawing planar partitions II: HH-drawings. In: Hromkovič, J., Sýkora, O. (eds.) WG 1998. LNCS, vol. 1517, pp. 124–136. Springer, Heidelberg (1998). CrossRefGoogle Scholar
  6. 6.
    Bishnu, A., Ghosh, A., Mathew, R., Mishra, G., Paul, S.: Grid obstacle representations of graphs (2017). coRR report arXiv:1708.01765
  7. 7.
    Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discrete Math. 86(1–3), 165–177 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Crescenzi, P., Di Battista, G., Piperno, A.: A note on optimal area algorithms for upward drawings of binary trees. Comput. Geom. 2, 187–200 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dujmovic, V., Morin, P.: On obstacle numbers. Electr. J. Comb. 22(3), P3.1 (2015)Google Scholar
  10. 10.
    Gewali, L., Ntafos, S.C.: Covering grids and orthogonal polygons with periscope guards. Comput. Geom. 2, 309–334 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Motwani, R., Raghunathan, A., Saran, H.: Covering orthogonal polygons with star polygons: the perfect graph approach. J. Comput. Syst. Sci. 40(1), 19–48 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Pach, J.: Graphs with no grid obstacle representation. Geombinatorics 26(2), 80–83 (2016)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Rosenstiehl, P., Tarjan, R.E.: Rectilinear planar layouts and bipolar orientation of planar graphs. Discrete Comput. Geom. 1, 343–353 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Tamassia, R., Tollis, I.: A unified approach to visibility representations of planar graphs. Disc. Comput. Geom. 1, 321–341 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Wismath, S.: Characterizing bar line-of-sight graphs. In: ACM Symposium on Computational Geometry (SoCG 1985), pp. 147–152. ACM (1985)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada

Personalised recommendations