Planarization and Acyclic Colorings of Subcubic Claw-Free Graphs

  • Christine Cheng
  • Eric McDermid
  • Ichiro Suzuki
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6986)


We study methods of planarizing and acyclically coloring claw-free subcubic graphs. We give a polynomial-time algorithm that, given such a graph G, produces an independent set Q of at most n/6 vertices whose removal from G leaves an induced planar subgraph P (in fact, P has treewidth at most four). We further show the stronger result that in polynomial-time a set of at most n/6 edges can be identified whose removal leaves a planar subgraph (of treewidth at most four). From an approximability point of view, we show that our results imply 6/5- and 9/8-approximation algorithms, respectively, for the (NP-hard) problems of finding a maximum induced planar subgraph and a maximum planar subgraph of a subcubic claw-free graph, respectively.

Regarding acyclic colorings, we give a polynomial-time algorithm that finds an optimal acyclic vertex coloring of a subcubic claw-free graph. To our knowledge, this represents the largest known subclass of subcubic graphs such that an optimal acyclic vertex coloring can be found in polynomial-time. We show that this bound is tight by proving that the problem is NP-hard for cubic line graphs (and therefore, claw-free graphs) of maximum degree d ≥ 4. An interesting corollary to the algorithm that we present is that there are exactly three subcubic claw-free graphs that require four colors to be acyclically colored. For all other such graphs, three colors suffice.


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  1. 1.
    Alon, N., McDiarmid, C., Reed, B.: Acyclic colourings of graphs. Random Structures and Algorithms 2, 277–288 (1990)CrossRefMATHGoogle Scholar
  2. 2.
    Alon, N., Zaks, A.: Algorithmic aspects of acyclic edge colorings. Algorithmica 32, 611–614 (2002)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Burnstein, M.I.: Every 4-valent graph has an acyclic five coloring, Soobšč. Akad. Nauk Gruzin SSR 93, 21–24 (1979) (in Russian)MathSciNetGoogle Scholar
  4. 4.
    Cǎlinescu, G., Fernandes, C.G., Finkler, U., Karloff, H.: A better approximation algorithm for finding planar subgraphs. Journal of Algorithms 27, 269–302 (1998)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Chudnovsky, M., Seymour, P.: The structure of claw-free graphs. In: Proceedings of the 20th British Combinatorial Conference, Surveys in Combinatorics 2005, Durham, pp. 153–171 (2005)Google Scholar
  6. 6.
    Edwards, K., Farr, G.: An Algorithm for Finding Large Induced Planar Subgraphs. In: Mutzel, P., Jünger, M., Leipert, S. (eds.) GD 2001. LNCS, vol. 2265, pp. 75–83. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  7. 7.
    Edwards, K., Farr, G.: Planarization and fragmentability of some classes of graphs. Discrete Mathematics 308, 2396–2406 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Faria, L., de Figueiredo, C.M.H., Mendonça, C.F.X.: Splitting number is NP-complete. In: Hromkovič, J., Sýkora, O. (eds.) WG 1998. LNCS, vol. 1517, pp. 285–297. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  9. 9.
    Faria, L., de Figueiredo, C.M.H., de Mendonça Neto, C.F.X.: On the complexity of the approximation of nonplanarity parameters for cubic graphs. Discrete Applied Mathematics 141(1-3), 119–134 (2004)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Faria, L., de Figueiredo, C.M.H., Gravier, S., de Mendonça Neto, C.F.X., Stolfi, J.: On maximum planar induced subgraphs. Discrete Applied Mathematics 154(13), 1774–1782 (2006)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gebremedhin, A.H., Manne, F., Pothen, A.: What color is your Jacobian? Graph coloring for computing derivatives. SIAM Review 47, 629–705 (2005)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kostochka, A., Stocker, C.: Graphs with maximum degree 5 are acyclically 7-colorable. Ars Mathematica Contemporanea 4, 153–164 (2011)MathSciNetMATHGoogle Scholar
  13. 13.
    Liebers, A.: Planarizing graphs – a survey and annotated bibliography. Journal of Graph Algorithms and Applications 5(1), 1–74 (2001)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Liu, P.C., Geldmacher, R.C.: On the deletion of nonplanar edges of a graph. Cong. Numer. 24, 727–738 (1979)MathSciNetMATHGoogle Scholar
  15. 15.
    Skulrattanakulchai, S.: Acyclic colorings of subcubic graphs. Information Processing Letters 92(4), 161–167 (2004)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Yannakakis, M.: Node and edge-deletion NP-complete problems. In: Proceedings of the 10th Annual ACM Symposium on Theory of Computing (STOC 1978), pp. 253–264 (1978)Google Scholar
  17. 17.
    Yannakakis, M.: Edge-deletion problems. SIAM J. Comput. 10, 297–309 (1981)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Zhang, X.-D., Bylka, S.: Disjoint triangles of a cubic line graph. Graphs and Combinatorics 20, 275–280 (2004)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Christine Cheng
    • 1
  • Eric McDermid
    • 1
  • Ichiro Suzuki
    • 1
  1. 1.Department of Computer ScienceUniversity of Wisconsin–MilwaukeeMilwaukeeUSA

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