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On Rectangle Intersection Graphs with Stab Number at Most Two

  • Dibyayan ChakrabortyEmail author
  • Sandip Das
  • Mathew C. Francis
  • Sagnik Sen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11394)

Abstract

Rectangle intersection graphs are the intersection graphs of axis-parallel rectangles in the plane. A graph G is said to be a k-stabbable rectangle intersection graph, or k-SRIG for short, if it has a rectangle intersection representation in which k horizontal lines can be placed such that each rectangle intersects at least one of them. The stab number of a graph G, denoted by stab(G), is the minimum integer k such that G is a k-SRIG. In this paper, we introduce some natural subclasses of 2-SRIG and study the containment relationships among them. We also give a linear time recognition algorithm for one of those classes. In this paper, we prove that the Chromatic Number problem is NP-complete even for 2-SRIGs. This strengthens a result by Imai and Asano [13]. We also show that triangle-free 2-SRIGs are three colorable.

References

  1. 1.
    Asinowski, A., Cohen, E., Golumbic, M.C., Limouzy, V., Lipshteyn, M., Stern, M.: Vertex intersection graphs of paths on a grid. J. Graph Algorithms Appl. 16(2), 129–150 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Benzer, S.: On the topology of the genetic fine structure. Proc. Natl. Acad. Sci. 45(11), 1607–1620 (1959)CrossRefGoogle Scholar
  3. 3.
    Bhore, S.K., Chakraborty, D., Das, S., Sen, S.: On a special class of boxicity 2 graphs. In: Ganguly, S., Krishnamurti, R. (eds.) CALDAM 2015. LNCS, vol. 8959, pp. 157–168. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-14974-5_16CrossRefGoogle Scholar
  4. 4.
    Bhore, S., Chakraborty, D., Das, S., Sen, S.: On local structures of cubicity 2 graphs. In: Chan, T.-H.H., Li, M., Wang, L. (eds.) COCOA 2016. LNCS, vol. 10043, pp. 254–269. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-48749-6_19CrossRefGoogle Scholar
  5. 5.
    Bogart, K.P., West, D.B.: A short proof that ‘proper = unit’. Discret. Math. 201(1–3), 21–23 (1999)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chakraborty, D., Francis, M.C., On the stab number of rectangle intersection graphs. CoRR, abs/1804.06571 (2018)Google Scholar
  7. 7.
    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).  https://doi.org/10.1007/978-3-642-25870-1_29CrossRefzbMATHGoogle Scholar
  8. 8.
    Corneil, D.G., Olariu, S., Stewart, L.: The LBFS structure and recognition of interval graphs. SIAM J. Discret. Math. 23(4), 1905–1953 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Cornelsen, S., Schank, T., Wagner, D.: Drawing graphs on two and three lines. In: Goodrich, M.T., Kobourov, S.G. (eds.) GD 2002. LNCS, vol. 2528, pp. 31–41. Springer, Heidelberg (2002).  https://doi.org/10.1007/3-540-36151-0_4CrossRefzbMATHGoogle Scholar
  10. 10.
    Cygan, M., et al.: Parameterized Algorithms, vol. 3. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-21275-3CrossRefzbMATHGoogle Scholar
  11. 11.
    Dvořák, Z., Kawarabayashi, K., Thomas, R.: Three-coloring triangle-free planar graphs in linear time. ACM Trans. Algorithms 7(4), 41:1–41:14 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Garey, M.R., Johnson, D.S., Miller, G.L., Papadimitriou, C.H.: The complexity of coloring circular arcs and chords. SIAM J. Algebr. Discret. Methods 1(2), 216–227 (1980)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Imai, H., Asano, T.: Finding the connected components and a maximum clique of an intersection graph of rectangles in the plane. J. Algorithms 4(4), 310–323 (1983)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kleinberg, J., Tardos, E.: Algorithm Design. Pearson Education India, Bangalore (2006)Google Scholar
  15. 15.
    Kratochvíl, J.: A special planar satisfiability problem and a consequence of its NP-completeness. Discret. Appl. Math. 52(3), 233–252 (1994)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lekkerkerker, C., Boland, J.: Representation of a finite graph by a set of intervals on the real line. Fundamenta Mathematicae 51(1), 45–64 (1962)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Perepelitsa, I.G.: Bounds on the chromatic number of intersection graphs of sets in the plane. Discret. Math. 262(1–3), 221–227 (2003)MathSciNetCrossRefGoogle Scholar
  18. 18.
    West, D.B.: Introduction to Graph Theory, 2nd edn. Prentice Hall, Upper Saddle River (2000)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Dibyayan Chakraborty
    • 1
    Email author
  • Sandip Das
    • 1
  • Mathew C. Francis
    • 2
  • Sagnik Sen
    • 3
  1. 1.Indian Statistical InstituteKolkataIndia
  2. 2.Indian Statistical InstituteChennaiIndia
  3. 3.Ramakrishna Mission Vivekananda Educational and Research InstituteHowrahIndia

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