, Volume 80, Issue 3, pp 977–994 | Cite as

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
Part of the following topical collections:
  1. Special Issue on Theoretical Informatics


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 graphs, complete bipartite graphs, multipartite graphs, bounded degree graphs, and genus-1 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-splittability in linear time, for a constant k.


Planarity Splittable Thickness Graph drawing Graph theory Complete graphs Genus-1 graphs NP-hardness Approximation Fixed-parameter tractable 



Most of the results of this paper were obtained at the McGill-INRIA-UVictoria 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. Can. J. Math. 14(17), 850–859 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Beineke, L.W., Harary, F.: A simplified NP-complete satisfiability problem. Craig A. Tovey 8, 85–89 (1984)MathSciNetGoogle Scholar
  3. 3.
    Borradaile, G., Eppstein, D., Zhu, P.: Planar induced subgraphs of sparse graphs. In: Duncan, C.A., Symvonis, A. (eds.) Proceedings of the 22nd International Symposium on Graph Drawing (GD). Lecture Notes Comput. Sci., vol. 8871, pp. 1–12. Springer (2014)Google Scholar
  4. 4.
    Chimani, M., Derka, M., Hliněnỳ, P., Klusáček, M.: How not to characterize planar-emulable graphs. Adv. Appl. Math. 50(1), 46–68 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Courcelle, B.: The monadic second-order logic of graphs. I. recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Courcelle, B.: On the expression of graph properties in some fragments of monadic second-order logic. In: Immerman, N., Kolaitis, P.G. (eds.) Proceedings of the DIMACS Workshop on Descriptive Complexity and Finite Models. DIMACS Ser. Discrete Math. Theor. Comput. Sci., vol. 31, pp. 33–62. American Math. Soc. (1996)Google Scholar
  7. 7.
    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
  8. 8.
    Dujmovic, V., Wood, D.R.: Graph treewidth and geometric thickness parameters. Discret. Comput. Geom. 37(4), 641–670 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    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), pp. 340–346. ACM (2004)Google Scholar
  10. 10.
    Eppstein, D., Kindermann, P., Kobourov, S.G., Liotta, G., Lubiw, A., Maignan, A., Mondal, D., Vosoughpour, H., Whitesides, S., Wismath, S.K.: On the planar split thickness of graphs. In: Kranakis, E., Navarro, G., Chávez, E. (eds.) Proceedings of the 12th Latin American Theoretical Informatics Symposium (LATIN). Lecture Notes Comput. Sci., vol. 9644, pp. 403–415. Springer (2016)Google Scholar
  11. 11.
    Faria, L., de Figueiredo, C.M.H., de Mendonça Neto, C.F.X.: Splitting number is NP-complete. Discret. Appl. Math. 108(1–2), 65–83 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fellows, M.R.: Encoding graphs in graphs. Ph.D. thesis, Univ. of California, San Diego (1985)Google Scholar
  13. 13.
    Gabow, H.N., Westermann, H.H.: Forests, frames, and games: algorithms for matroid sums and applications. Algorithmica 7(1), 465–497 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gansner, E.R., Hu, Y., Kobourov, S.G.: Gmap: Visualizing graphs and clusters as maps. In: Proceedings of the IEEE Pacific Visualization Symposium (PacificVis), pp. 201–208 (2010)Google Scholar
  15. 15.
    Hartsfield, N.: The toroidal splitting number of the complete graph \({K}_n\). Discrete Math. (1986)Google Scholar
  16. 16.
    Hartsfield, N.: The splitting number of the complete graph in the projective plane. Graphs Comb. 3(1), 349–356 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hartsfield, N., Jackson, B., Ringel, G.: The splitting number of the complete graph. Graphs Comb. 1(1), 311–329 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Heawood, P.J.: Map colour theorem. Q. J. Math. 24, 332–338 (1890)zbMATHGoogle Scholar
  19. 19.
    Huneke, J.P.: A conjecture in topological graph theory. In: Robertson, N., Seymour, P.D. (eds.) Graph Structure Theory, Proceedings of a AMS-IMS-SIAM Joint Summer Research Conference on Graph Minors held June 22 to July 5, 1991, at the University of Washington, Seattle. Contemp. Math., vol. 147, pp. 387–389. American Math. Soc. (1991)Google Scholar
  20. 20.
    Hutchinson, J.P.: Coloring ordinary maps, maps of empires, and maps of the moon. Math. Mag. 66(4), 211–226 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Jackson, B., Ringel, G.: The splitting number of complete bipartite graphs. Arch. Math. 42, 178–184 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Jackson, B., Ringel, G.: Splittings of graphs on surfaces. In: Harary, F., Maybee, J.S. (eds.) Proceedings of the 1st Colorado Symposium on Graph Theory, pp. 203–219 (1985)Google Scholar
  23. 23.
    Knauer, K., Ueckerdt, T.: Three ways to cover a graph. Discret. Math. 339(2), 745–758 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kratochvíl, J., Lubiw, A., Nesetril, J.: Noncrossing subgraphs in topological layouts. SIAM J. Discret. Math. 4(2), 223–244 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Liebers, A.: Planarizing graphs—a survey and annotated bibliography. J. Graph Algorithms Appl. 5(1), 1–74 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Nash-Williams, C.S.J.A.: Decomposition of finite graphs into forests. J. Lond. Math. Soc. 39(1), 12 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Negami, S.: Enumeration of projective-planar embeddings of graphs. Discret. Math. 63(3), 299–306 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Negami, S.: The spherical genus and virtually planar graphs. Discret. Math. 70(2), 159–168 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Picard, J.C., Queyranne, M.: A network flow solution to some nonlinear 0–1 programming problems, with applications to graph theory. Networks 12(2), 141–159 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Riche, N.H., Dwyer, T.: Untangling euler diagrams. Proc. IEEE Trans. Vis. Comput. Graph. 16(6), 1090–1099 (2010)CrossRefGoogle Scholar
  32. 32.
    Ringel, G., Jackson, B.: Solution of Heawood’s empire problem in the plane. J. Reine Angew. Math. 347, 146–153 (1984)MathSciNetzbMATHGoogle Scholar
  33. 33.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Thomason, A.: The extremal function for complete minors. J. Comb. Theory Ser. B 81(2), 318–338 (2001)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  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.Universit. Grenoble AlpesGrenobleFrance
  7. 7.University of VictoriaVictoriaCanada
  8. 8.University of LethbridgeLethbridgeCanada

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