Advertisement

Bend-Bounded Path Intersection Graphs: Sausages, Noodles, and Waffles on a Grill

  • Steven Chaplick
  • Vít Jelínek
  • Jan Kratochvíl
  • Tomáš Vyskočil
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7551)

Abstract

In this paper we study properties of intersection graphs of k-bend paths in the rectangular grid. A k-bend path is a path with at most k 90 degree turns. The class of graphs representable by intersections of k-bend paths is denoted by B k -VPG. We show here that for every fixed k, B k -VPG \(\subsetneq\) B k + 1-VPG and that recognition of graphs from B k -VPG is NP-complete even when the input graph is given by a B k + 1-VPG representation. We also show that the class B k -VPG (for k ≥ 1) is in no inclusion relation with the class of intersection graphs of straight line segments in the plane.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Asinowski, A., Cohen, E., Golumbic, M.C., Limouzy, V., Lipshteyn, M., Stern, M.: String graphs of k-bend paths on a grid. Electronic Notes in Discrete Mathematics 37, 141–146 (2011)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Asinowski, A., Cohen, E., Golumbic, M.C., Limouzy, V., Lipshteyn, M., Stern, M.: Vertex Intersection Graphs of Paths on a Grid. Journal of Graph Algorithms and Applications 16(2), 129–150 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bandy, M., Sarrafzadeh, M.: Stretching a knock-knee layout for multilayer wiring. IEEE Trans. Computing 39, 148–151 (1990)CrossRefGoogle Scholar
  4. 4.
    Bellantoni, S., Ben-Arroyo Hartman, I., Przytycka, T.M., Whitesides, S.: Grid intersection graphs and boxicity. Discrete Mathematics 114, 41–49 (1993)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Chaplick, S., Cohen, E., Stacho, J.: Recognizing Some Subclasses of Vertex Intersection Graphs of 0-Bend Paths in a Grid. In: Kolman, P., Kratochvíl, J. (eds.) WG 2011. LNCS, vol. 6986, pp. 319–330. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  6. 6.
    Coury, M.D., Hell, P., Kratochvíl, J., Vyskočil, T.: Faithful Representations of Graphs by Islands in the Extended Grid. In: López-Ortiz, A. (ed.) LATIN 2010. LNCS, vol. 6034, pp. 131–142. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  7. 7.
    Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing. Prentice-Hall (1999)Google Scholar
  8. 8.
    Golumbic, M.C., Ries, B.: On the intersection graphs of orthogonal line segments in the plane: characterizations of some subclasses of chordal graphs. To Appear in Graphs and CombinatoricsGoogle Scholar
  9. 9.
    Kratochvíl, J.: String graphs II, Recognizing string graphs is NP-hard. J. Comb. Theory, Ser. B 52, 67–78 (1991)MATHCrossRefGoogle Scholar
  10. 10.
    Kratochvíl, J., Matoušek, J.: String graphs requiring exponential representations. J. Comb. Theory, Ser. B 53, 1–4 (1991)MATHCrossRefGoogle Scholar
  11. 11.
    Kratochvíl, J.: A Special Planar Satisfiability Problem and a Consequence of Its NP-completeness. Discrete Applied Mathematics 52, 233–252 (1994)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Kratochvíl, J., Matoušek, J.: Intersection Graphs of Segments. J. Comb. Theory, Ser. B 62, 289–315 (1994)MATHCrossRefGoogle Scholar
  13. 13.
    Molitor, P.: A survey on wiring. EIK Journal of Information Processing and Cybernetics 27, 3–19 (1991)MATHGoogle Scholar
  14. 14.
    Schaefer, M., Sedgwick, E., Stefankovic, D.: Recognizing string graphs in NP. J. Comput. Syst. Sci. 67, 365–380 (2003)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Sinden, F.: Topology of thin film circuits. Bell System Tech. J. 45, 1639–1662 (1966)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Steven Chaplick
    • 1
  • Vít Jelínek
    • 2
  • Jan Kratochvíl
    • 3
  • Tomáš Vyskočil
    • 3
  1. 1.Department of MathematicsWilfrid Laurier UniversityWaterlooCanada
  2. 2.Computer Science Institute, Faculty of Mathematics and PhysicsCharles UniversityPragueCzech Republic
  3. 3.Department of Applied Mathematics, Faculty of Mathematics and PhysicsCharles UniversityPragueCzech Republic

Personalised recommendations