On Split \(B_1\)-EPG Graphs

  • Zakir Deniz
  • Simon Nivelle
  • Bernard Ries
  • David Schindl
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)


In this paper, we are interested in edge intersection graphs of paths in a grid, such that each such path has at most one bend. These graphs were introduced in [12] and they are called \(B_1\)-EPG graphs. In particular, we focus on split graphs and characterise those that are \(B_1\)-EPG. This characterisation allows us to disprove a conjecture of Cameron et al. [7]. The existence of polynomial-time recognition algorithm for this graph class is still unknown. We furthermore investigate inclusion relationships among subclasses of split graphs that are \(B_1\)-EPG.



The first author has been supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK) under 2214-A Doctoral Research Program Grant during his stay in Switzerland. This work was done while the first and second authors visited the University of Fribourg, Switzerland. The support of the institution is gratefully acknowledged.


  1. 1.
    Alcón, L., Bonomo, F., Durán, G., Gutierrez, M., Mazzoleni, M.P., Ries, B., Valencia-Pabon, M.: On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid. Discrete Appl. Math. 234, 12–21 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Asinowski, A., Ries, B.: Some properties of edge intersection graphs of single-bend paths on a grid. Discrete Math. 312, 427–440 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Asinowski, A., Suk, A.: Edge intersection graphs of systems of paths on a grid with a bounded number of bends. Discrete Appl. Math. 157(14), 3174–3180 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bield, T., Stern, M.: Edge-intersection graphs of \(k\)-bend paths in grids. Discrete Math. Theor. Comput. Sci. 12:1, 1–12 (2010)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bonomo, F., Mazzoleni, M.P., Stein, M.: Clique coloring \({B}_1\)-EPG graphs. Discrete Math. 340(5), 1008–1011 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bougeret, M., Bessy, S., Gonçalves, D., Paul, C.: On independent set on B1-EPG graphs. In: Sanità, L., Skutella, M. (eds.) WAOA 2015. LNCS, vol. 9499, pp. 158–169. Springer, Cham (2015). CrossRefGoogle Scholar
  7. 7.
    Cameron, K., Chaplick, S., Hoang, C.T.: Edge intersection graphs of L-shaped paths in grids. Discrete Appl. Math. 210, 185–194 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cohen, E., Golumbic, M.C., Ries, B.: Characterizations of cographs as intersection graphs of paths on a grid. Discrete Appl. Math. 178, 46–57 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Diestel, R.: Graph Theory. Springer, Heidelberg (2005)zbMATHGoogle Scholar
  10. 10.
    Epstein, D., Golumbic, M.C., Morgenstern, G.: Approximation algorithms for \({B}_1\)-EPG graphs. In: Dehne, F., Solis-Oba, R., Sack, J.-R. (eds.) WADS 2013. LNCS, vol. 8037, pp. 328–340. Springer, Heidelberg (2013). CrossRefGoogle Scholar
  11. 11.
    Francis, M.C., Lahiri, A.: VPG and EPG bend-numbers of halin graphs. Discrete Appl. Math. 215, 95–105 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Golumbic, M., Lipshteyn, M., Stern, M.: Edge intersection graphs of single bend paths on a grid. Networks 54, 130–138 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Heldt, D., Knauer, K., Ueckerdt, T.: Edge-intersection graphs of grid paths: the bend-number. Discrete Appl. Math. 167, 144–162 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Heldt, D., Knauer, K., Ueckerdt, T.: On the bend-number of planar and outerplanar graphs. Discrete Appl. Math. 179, 109–119 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Pergel, M., Rzążewski, P.: On edge intersection graphs of paths with 2 bends. In: Heggernes, P. (ed.) WG 2016. LNCS, vol. 9941, pp. 207–219. Springer, Heidelberg (2016). CrossRefGoogle Scholar
  16. 16.
    West, D.B.: Introduction to Graph Theory. Prentice-Hall, Upper Saddle River (1996)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Duzce UniversityDuzceTurkey
  2. 2.ENS Paris SaclayParisFrance
  3. 3.University of FribourgFribourgSwitzerland
  4. 4.Geneva School of Business AdministrationGenevaSwitzerland

Personalised recommendations