Journal of Combinatorial Optimization

, Volume 33, Issue 4, pp 1183–1225 | Cite as

A tighter insertion-based approximation of the crossing number

  • Markus Chimani
  • Petr HliněnýEmail author


Let G be a planar graph and F a set of additional edges not yet in G. The multiple edge insertion problem (MEI) asks for a drawing of \(G+F\) with the minimum number of pairwise edge crossings, such that the subdrawing of G is plane. Finding an exact solution to MEI is NP-hard for general F. We present the first polynomial time algorithm for MEI that achieves an additive approximation guarantee—depending only on the size of F and the maximum degree of G, in the case of connected G. Our algorithm seems to be the first directly implementable one in that realm, too, next to the single edge insertion. It is also known that an (even approximate) solution to the MEI problem would approximate the crossing number of the F-almost-planar graph \(G+F\), while computing the crossing number of \(G+F\) exactly is NP-hard already when \(|F|=1\). Hence our algorithm induces new, improved approximation bounds for the crossing number problem of F-almost-planar graphs, achieving constant-factor approximation for the large class of such graphs of bounded degrees and bounded size of F.


Planar graph Multiple edge insertion SPQR tree Crossing number 


  1. Bhatt SN, Leighton FT (1984) A framework for solving vlsi graph layout problems. J Comput Syst Sci 28(2):300–343MathSciNetCrossRefzbMATHGoogle Scholar
  2. Bienstock D, Monma CL (1990) On the complexity of embedding planar graphs to minimize certain distance measures. Algorithmica 5(1):93–109MathSciNetCrossRefzbMATHGoogle Scholar
  3. Cabello S, Mohar B (2011) Crossing and weighted crossing number of near planar graphs. Algorithmica 60:484–504MathSciNetCrossRefzbMATHGoogle Scholar
  4. Cabello S, Mohar B (2013) Adding one edge to planar graphs makes crossing number and 1-planarity hard. SIAM J Comput 42:1803–1829MathSciNetCrossRefzbMATHGoogle Scholar
  5. Chimani M (2008) Computing crossing numbers. PhD thesis, TU Dortmund, Germany, Online.
  6. Chimani M, Gutwenger C (2012) Advances in the planarization method: effective multiple edge insertions. J. Graph Algorithms Appl 16(3):729–757MathSciNetCrossRefzbMATHGoogle Scholar
  7. Chimani M, Gutwenger C, Mutzel P, Wolf C (2009) Inserting a vertex into a planar graph. In: Proceedings SODA ’09, pp 375–383Google Scholar
  8. Chimani M, Hliněný P (2011) A tighter insertion-based approximation of the crossing number. In: Proceedngs ICALP ’11, vol 6755 of LNCS, Springer, New York, pp 122–134Google Scholar
  9. Chimani M, Hliněný P (2016) Inserting multiple edges into a planar graph. In: Proceedings SoCG ’16, pages—to appear. Dagstuhl, arXiv:1509.07952
  10. Chimani M, Hliněný P, Mutzel P (2012) Vertex insertion approximates the crossing number for apex graphs. Eur J Comb 33:326–335MathSciNetCrossRefzbMATHGoogle Scholar
  11. Chuzhoy J (2011) An algorithm for the graph crossing number problem. In Proceedings STOC ’11, ACM, pp 303–312Google Scholar
  12. Chuzhoy J, Makarychev Y, Sidiropoulos A (2011) On graph crossing number and edge planarization. In: Proceedings SODA ’11. ACM Press, New York, pp 1050–1069Google Scholar
  13. Di Battista G, Tamassia R (1996) On-line planarity testing. SIAM J Comput 25:956–997MathSciNetCrossRefzbMATHGoogle Scholar
  14. Even G, Guha S, Schieber B (2002) Improved approximations of crossings in graph drawings and VLSI layout areas. SIAM J Comput 32(1):231–252MathSciNetCrossRefzbMATHGoogle Scholar
  15. Gitler I, Hliněný P, Leanos J, Salazar G (2008) The crossing number of a projective graph is quadratic in the face-width. Electron J Comb 15(1):46Google Scholar
  16. Gutwenger C (2010) Application of SPQR-trees in the planarization approach for drawing graphs. PhD thesis, TU Dortmund, GermanyGoogle Scholar
  17. Gutwenger C, Mutzel P (2001) A linear time implementation of SPQR trees. In: Proceedings GD ’00, vol 1984 of LNCS. Springer, New York, pp 77–90Google Scholar
  18. Gutwenger C, Mutzel P (2004) An experimental study of crossing minimization heuristics. In: Proceedings GD ’03, vol 2912. LNCS. Springer, New York, pp 13–24Google Scholar
  19. Gutwenger C, Mutzel P, Weiskircher R (2005) Inserting an edge into a planar graph. Algorithmica 41(4):289–308MathSciNetCrossRefzbMATHGoogle Scholar
  20. Hliněný P, Chimani M (2010) Approximating the crossing number of graphs embeddable in any orientable surface. In: Proceedings SODA ’10, pp 918–927Google Scholar
  21. Hliněný P, Salazar G (2006) On the crossing number of almost planar graphs. In: Proceedings GD ’05, vol 4372. LNCS. Springer, New York, pp 162–173Google Scholar
  22. Hliněný P, Salazar G (2007) Approximating the crossing number of toroidal graphs. In Proceedings ISAAC ’07, vol 4835. LNCS. Springer, New York, pp 148–159Google Scholar
  23. Hopcroft JE, Tarjan RE (1973) Dividing a graph into triconnected components. SIAM J Comput 2(3):135–158MathSciNetCrossRefzbMATHGoogle Scholar
  24. Masuda S, Nakajima K, Kashiwabara T, Fujisawa T (1990) Crossing minimization in linear embeddings of graphs. IEEE Trans Comput 39:124–127MathSciNetCrossRefGoogle Scholar
  25. Tutte WT (1966) Connectivity in graphs. In: Mathematical expositions, vol 15. University of Toronto Press, TorontoGoogle Scholar
  26. Vrt’o I (2014) Crossing numbers of graphs: a bibliography.
  27. Ziegler T (2001) Crossing minimization in automatic graph drawing. PhD thesis, Saarland University, GermanyGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Theoretical Computer ScienceUniversity OsnabrückOsnabrückGermany
  2. 2.Faculty of InformaticsMasaryk UniversityBrnoCzech Republic

Personalised recommendations