Advertisement

Bounds on the Bend Number of Split and Cocomparability Graphs

  • Dibyayan Chakraborty
  • Sandip Das
  • Joydeep MukherjeeEmail author
  • Uma Kant Sahoo
Article

Abstract

A path is a simple, piecewise linear curve made up of alternating horizontal and vertical line segments in the plane. A k-bend path is a path made up of at most k + 1 line segments. A Bk-VPG representation of a graph is a collection of k-bend paths such that each path in the collection represents a vertex of the graph and two such paths intersect if and only if the vertices they represent are adjacent in the graph. The graphs that have a Bk-VPG representation are calledBk-VPG graphs. The bend number of a graph G, denoted by bend(G), is the minimum integer k for which G has a Bk-VPG representation. In this paper, we study the relationship between poset dimension and bend number of cocomparability graphs. It is known that the poset dimension dim(G) of a cocomparability graph G is greater than or equal to its bend number bend(G). Cohen et al. (order2016) asked for examples of cocomparability graphs with low bend number and high poset dimension. We answer this question by proving that for each \(m, t \in \mathbb {N}\), there exists a cocomparability graph Gt, m with t < bend(Gt, m) ≤ 4t + 29 and dim(Gt, m) − bend(Gt, m) > m. The techniques used to prove the above result allow us to partially address the open question posed by Chaplick et al. (wg2012) who asked whether Bk-VPG-chordal \(\subsetneq ~B_{k + 1}\)-VPG-chordal for all \(k \in \mathbb {N}\). We address this by proving that there are infinitely many \(m \in \mathbb {N}\) such that Bm-VPG-split \(\subsetneq ~B_{m + 1}\)-VPG-split which provides infinitely many positive examples. We use the same techniques to prove that, for all \(t \in \mathbb {N}\), Bt-VPG-\(Forb(C_{\geq 5})~\subsetneq ~B_{4t + 29}\)-VPG-Forb(C≥ 5), where Forb(C≥ 5) denotes the family of graphs that does not contain induced cycles of length greater than 4. Furthermore, we show that for all \(t \in \mathbb {N}\), PBt-VPG-split \(\subsetneq PB_{36t + 80}\)-VPG-split, where PBt-VPG denotes the class of graphs with proper bend number at most t (i.e. it has a Bt-VPG representation in which two paths have only finitely many intersection points, each intersection point belongs to exactly two paths, and whenever two paths intersect they cross each other).

Keywords

String graphs Split graphs Cocomparability graphs Bk-VPG graphs Bend number Poset dimension 

Notes

Acknowledgements

We thank Sagnik Sen for helpful comments in preparing the manuscript. We thank Subhodeep Ranjan Dev for carefully reading the final draft. We would like to thank the anonymous referees for meticulously reading the manuscript, and for helpful suggestions which made the proofs more rigorous and increased the readability and flow of the manuscript. Joydeep Mukherjee is supported by DST SERB NPDF fellowship (PDF/2016/001647).

References

  1. 1.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bertossi, A.A.: Dominating sets for split and bipartite graphs. Inf. Process. Lett. 19(1), 37–40 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Biedl, T.C., Derka, M.: 1-String B 2-VPG representation of planar graphs. In: 31St International Symposium on Computational Geometry, SoCG 2015, June 22-25, 2015, Eindhoven, The Netherlands, pp. 141–155 (2015)Google Scholar
  4. 4.
    Chaplick, S., Cohen, E., Stacho, J.: Recognizing some subclasses of vertex intersection graphs of 0-bend paths in a grid. In: Graph-Theoretic Concepts in Computer Science, pp. 319–330. Springer (2011)Google Scholar
  5. 5.
    Chaplick, S., Jelínek, V., Kratochvil, J., Vyskočil, T.: Bend-bounded path intersection graphs: sausages, noodles, and waffles on a grill. In: Graph-Theoretic Concepts in Computer Science, pp. 274–285. Springer (2012)Google Scholar
  6. 6.
    Cohen, E., Golumbic, M.C., Trotter, W.T., Wang, R.: Posets and vpg graphs. Order 33(1), 39–49 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fox, J., Pach, J.: String graphs and incomparability graphs. Adv. Math. 230(3), 1381–1401 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gavril, F.: The intersection graphs of subtrees in trees are exactly the chordal graphs. Journal of Combinatorial Theory, Series B 16(1), 47–56 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Golumbic, M.C., Rotem, D., Urrutia, J.: Comparability graphs and intersection graphs. Discrete Math. 43(1), 37–46 (1983).  https://doi.org/10.1016/0012-365X(83)90019-5 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gonçalves, D., Isenmann, L., Pennarun, C.: Planar graphs as l-intersection or l-contact graphs. In: Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’18, pp. 172–184. New Orleans, Louisiana (2018)Google Scholar
  11. 11.
    Harary, F.: Graph Theory. Addison Wesley, Boston (1969)CrossRefzbMATHGoogle Scholar
  12. 12.
    Kratochvíl, J.: String graphs. I. the number of critical nonstring graphs is infinite. Journal of Combinatorial Theory, Series B 52(1), 53–66 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kratochvíl, J.: String graphs. II. recognizing string graphs is NP-hard. Journal of Combinatorial Theory, Series B 52(1), 67–78 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kratochvíl, J., Matoušek, J.: String graphs requiring exponential representations. Journal of Combinatorial Theory, Series B 53(1), 1–4 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lahiri, A., Mukherjee, J., Subramanian, C.: Maximum independent Set on B 1-VPG Graphs. In: Combinatorial Optimization and Applications, pp. 633–646. Springer (2015)Google Scholar
  16. 16.
    Matoušek, J.: String graphs and separators. In: Geometry, Structure and Randomness in Combinatorics, pp. 61–97. Springer (2014)Google Scholar
  17. 17.
    Mehrabi, S.: Approximation algorithms for independence and domination on B 1-VPG and B 1-EPG graphs. arXiv:1702.05633 (2017)
  18. 18.
    Schaefer, M., Sedgwick, E., Štefankovič, D.: Recognizing string graphs in NP. J. Comput. Syst. Sci. 67(2), 365–380 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Sinden, F.W.: Topology of thin film rc circuits. Bell Syst. Tech. J. 45(9), 1639–1662 (1966)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Dibyayan Chakraborty
    • 1
  • Sandip Das
    • 1
  • Joydeep Mukherjee
    • 1
    Email author
  • Uma Kant Sahoo
    • 1
  1. 1.Indian Statistical InstituteKolkataIndia

Personalised recommendations