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Obstructing Visibilities with One Obstacle

  • Steven Chaplick
  • Fabian Lipp
  • Ji-won Park
  • Alexander Wolff
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9801)

Abstract

Obstacle representations of graphs have been investigated quite intensely over the last few years. We focus on graphs that can be represented by a single obstacle. Given a (topologically open) non-self-intersecting polygon C and a finite set P of points in general position in the complement of C, the visibility graph \(G_C(P)\) has a vertex for each point in P and an edge pq for any two points p and q in P that can see each other, that is, \(\overline{pq} \cap C=\emptyset \). We draw \(G_C(P)\) straight-line and call this a visibility drawing. Given a graph G, we want to compute an obstacle representation of G, that is, an obstacle C and a set of points P such that \(G=G_C(P)\). The complexity of this problem is open, even when the points are exactly the vertices of a simple polygon and the obstacle is the complement of the polygon—the simple-polygon visibility graph problem.

There are two types of obstacles; outside obstacles lie in the unbounded component of the visibility drawing, whereas inside obstacles lie in the complement of the unbounded component. We show that the class of graphs with an inside-obstacle representation is incomparable with the class of graphs that have an outside-obstacle representation. We further show that any graph with at most seven vertices has an outside-obstacle representation, which does not hold for a specific graph with eight vertices. Finally, we show NP-hardness of the outside-obstacle graph sandwich problem: given graphs G and H on the same vertex set, is there a graph K such that \(G \subseteq K \subseteq H\) and K has an outside-obstacle representation. Our proof also shows that the simple-polygon visibility graph sandwich problem, the inside-obstacle graph sandwich problem, and the single-obstacle graph sandwich problem are all NP-hard.

Keywords

Simple Polygon Visibility Graph Outerplanar Graph Unit Disk Graph Unbounded Component 
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.
    Alpert, H., Koch, C., Laison, J.D.: Obstacle numbers of graphs. Discrete Comput. Geom. 44(1), 223–244 (2009). http://dx.doi.org/10.1007/s00454-009-9233-8 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Balko, M., Cibulka, J., Valtr, P.: Drawing graphs using a small number of obstacles. In: Di Giacomo, E., Lubiw, A. (eds.) GD 2015. LNCS, vol. 9411, pp. 360–372. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-27261-0_30 CrossRefGoogle Scholar
  3. 3.
    Berman, L.W., Chappell, G.G., Faudree, J.R., Gimbel, J., Hartman, C., Williams, G.I.: Graphs with obstacle number greater than one. Arxiv report arXiv.org/abs/1606.03782 (2016)
  4. 4.
    Cardinal, J., Hoffmann, U.: Recognition and complexity of point visibility graphs. In: Arge, L., Pach, J. (eds.) Proceedings of the 31st International Symposium Computational Geometry (SoCG 2015), LIPIcs, vol. 34, pp. 171–185. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)Google Scholar
  5. 5.
    Chaplick, S., Lipp, F., Park, J.w., Wolff, A.: Obstructing visibilities with one obstacle. Arxiv report arXiv.org/abs/1607.00278v2 (2016)
  6. 6.
    Dujmović, V., Morin, P.: On obstacle numbers. Electr. J. Combin. 33(3), paper #P3.1, 7 p. (2015). arXiv.org/abs/1308.4321
  7. 7.
    Fulek, R., Saeedi, N., Sarıöz, D.: Convex obstacle numbers of outerplanar graphs and bipartite permutation graphs. In: Pach, J. (ed.) Thirty Essays on Geometric Graph Theory, pp. 249–261. Springer, New York (2013)CrossRefGoogle Scholar
  8. 8.
    Ghosh, S.K., Goswami, P.P.: Unsolved problems in visibility graphs of points, segments and polygons. Arxiv report arXiv.org/abs/1012.5187v4 (2012)
  9. 9.
    Golumbic, M.C., Kaplan, H., Shamir, R.: Graph sandwich problems. J. Algorithms 19(3), 449–473 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Johnson, M.P., Sarıöz, D.: Representing a planar straight-line graph using few obstacles. In: Proceedings of the 26th Canadian Conference Computational Geometry (CCCG 2014), pp. 95–99 (2014). http://www.cccg.ca/proceedings/2014/papers/paper14.pdf
  11. 11.
    Koch, A., Krug, M., Rutter, I.: Graphs with plane outside-obstacle representations. Arxiv report arXiv.org/abs/1306.2978 (2013)
  12. 12.
    Mukkamala, P., Pach, J., Pálvölgyi, D.: Lower bounds on the obstacle number of graphs. Electr. J. Combin. 19(2), paper #P32, 8 p. (2012). http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i2p32
  13. 13.
    Mukkamala, P., Pach, J., Sarıöz, D.: Graphs with large obstacle numbers. In: Thilikos, D.M. (ed.) WG 2010. LNCS, vol. 6410, pp. 292–303. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-16926-7_27 CrossRefGoogle Scholar
  14. 14.
    Pach, J., Sarıöz, D.: On the structure of graphs with low obstacle number. Graphs Comb. 27(3), 465–473 (2011). http://dx.doi.org/10.1007/s00373-011-1027-0 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of the 10th Annual ACM Symposium Theory Computing (STOC 1978), pp. 216–226 (1978). http://dx.doi.org/10.1145/800133.804350

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Steven Chaplick
    • 1
  • Fabian Lipp
    • 1
  • Ji-won Park
    • 2
  • Alexander Wolff
    • 1
  1. 1.Lehrstuhl für Informatik IUniversität WürzburgWürzburgGermany
  2. 2.KAISTDaejeonKorea

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