On the Planar Split Thickness of Graphs

  • David Eppstein
  • Philipp Kindermann
  • Stephen Kobourov
  • Giuseppe Liotta
  • Anna Lubiw
  • Aude Maignan
  • Debajyoti Mondal
  • Hamideh Vosoughpour
  • Sue Whitesides
  • Stephen Wismath
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9644)


Motivated by applications in graph drawing and information visualization, we examine the planar split thickness of a graph, that is, the smallest k such that the graph is k-splittable into a planar graph. A k-split operation substitutes a vertex v by at most k new vertices such that each neighbor of v is connected to at least one of the new vertices.

We first examine the planar split thickness of complete and complete bipartite graphs. We then prove that it is NP-hard to recognize graphs that are 2-splittable into a planar graph, and show that one can approximate the planar split thickness of a graph within a constant factor. If the treewidth is bounded, then we can even verify k-splittablity in linear time, for a constant k.



Most of the results of this paper were obtained at the McGill-INRIA-Victoria Workshop on Computational Geometry, Barbados, February 2015. We would like to thank the organizers of these events, as well as many participants for fruitful discussions and suggestions. The first, fourth, sixth, and eighth authors acknowledge the support from NSF grant 1228639, 2012C4E3KT PRIN Italian National Research Project, PEPS egalite project, and NSERC respectively.


  1. 1.
    Beineke, L.W., Harary, F.: The thickness of the complete graph. Canad. J. Math. 14(17), 850–859 (1965)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Borradaile, G., Eppstein, D., Zhu, P.: Planar induced subgraphs of sparse graphs. In: Duncan, C., Symvonis, A. (eds.) GD 2014. LNCS, vol. 8871, pp. 1–12. Springer, Heidelberg (2014)Google Scholar
  3. 3.
    Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inform. Comput. 85(1), 12–75 (1990)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Courcelle, B.: On the expression of graph properties in some fragments of monadic second-order logic. In: Immerman, N., Kolaitis, P.G. (eds.) Proc. Descr. Complex. Finite Models. DIMACS, vol. 31, pp. 33–62. Amer. Math. Soc. (1996)Google Scholar
  5. 5.
    Dujmovic, V., Wood, D.R.: Graph treewidth and geometric thickness parameters. Discrete Comput. Geom. 37(4), 641–670 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Duncan, C.A., Eppstein, D., Kobourov, S.G.: The geometric thickness of low degree graphs. In: Snoeyink, J., Boissonnat, J. (eds.) Proceedings of the 20th ACM Symposium on Computational Geometry (SOCG 2004). pp. 340–346. ACM (2004)Google Scholar
  7. 7.
    Faria, L., de Figueiredo, C.M.H., de Mendonça Neto, C.F.X.: Splitting number is NP-complete. Discrete Appl. Math. 108(1–2), 65–83 (2001)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hartsfield, N.: The toroidal splitting number of the complete graph \({K}_n\). Discrete Math. 62, 35–47 (1986)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hartsfield, N.: The splitting number of the complete graph in the projective plane. Graphs Comb. 3(1), 349–356 (1987)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hartsfield, N., Jackson, B., Ringel, G.: The splitting number of the complete graph. Graphs Comb. 1(1), 311–329 (1985)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Heawood, P.J.: Map colour theorem. Quart. J. Math. 24, 332–338 (1890)MATHGoogle Scholar
  12. 12.
    Hutchinson, J.P.: Coloring ordinary maps, maps of empires, and maps of the moon. Math. Mag. 66(4), 211–226 (1993)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Knauer, K., Ueckerdt, T.: Three ways to cover a graph. Arxiv report (2012). http://arxiv.org/abs/1205.1627
  14. 14.
    Kratochvíl, J., Lubiw, A., Nesetril, J.: Noncrossing subgraphs in topological layouts. SIAM J. Discrete Math. 4(2), 223–244 (1991)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Liebers, A.: Planarizing graphs - a survey and annotated bibliography. J. Graph Algor. Appl. 5(1), 1–74 (2001)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    de Mendonça Neto, C.F.X., Schaffer, K., Xavier, E.F., Stolfi, J., Faria, L., de Figueiredo, C.M.H.: The splitting number and skewness of \({C}_n\times {C}_m\). Ars Comb. 63 (2002)Google Scholar
  17. 17.
    Morgenstern, M.: Existence and explicit constructions of \(q+1\) regular Ramanujan graphs for every prime power \(q\). J. Comb. Theory, Ser. B 62(1), 44–62 (1994)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Nash-Williams, C.: Decomposition of finite graphs into forests. J. London Math. Soc. 39(1), 12 (1964)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Ringel, G., Jackson, B.: Solution of Heawood’s empire problem in the plane. J. Reine Angew. Math. 347, 146–153 (1984)MathSciNetMATHGoogle Scholar
  20. 20.
    Scheinerman, E.R., West, D.B.: The interval number of a planar graph: Three intervals suffice. J. Comb. Theory, Ser. B 35(3), 224–239 (1983)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Thomason, A.: The extremal function for complete minors. J. Comb. Theory, Ser. B 81(2), 318–338 (2001)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • David Eppstein
    • 1
  • Philipp Kindermann
    • 2
  • Stephen Kobourov
    • 3
  • Giuseppe Liotta
    • 4
  • Anna Lubiw
    • 5
  • Aude Maignan
    • 6
  • Debajyoti Mondal
    • 7
  • Hamideh Vosoughpour
    • 5
  • Sue Whitesides
    • 8
  • Stephen Wismath
    • 9
  1. 1.University of CaliforniaIrvineUSA
  2. 2.FernUniversität HagenHagenGermany
  3. 3.University of ArizonaTucsonUSA
  4. 4.Università Degli Studi di PerugiaPerugiaItaly
  5. 5.University of WaterlooWaterlooCanada
  6. 6.Université Grenoble AlpesGrenobleFrance
  7. 7.University of ManitobaWinnipegCanada
  8. 8.University of VictoriaVictoriaCanada
  9. 9.University of LethbridgeLethbridgeCanada

Personalised recommendations