Domination When the Stars Are Out

  • Danny Hermelin
  • Matthias Mnich
  • Erik Jan van Leeuwen
  • Gerhard J. Woeginger
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)


We algorithmize the recent structural characterization for claw-free graphs by Chudnovsky and Seymour. Building on this result, we show that Dominating Set on claw-free graphs is (i) fixed-parameter tractable and (ii) even possesses a polynomial kernel. To complement these results, we establish that Dominating Set is not fixed-parameter tractable on the slightly larger class of graphs that exclude K 1,4 as an induced subgraph. Our results provide a dichotomy for Dominating Set in K 1,ℓ-free graphs and show that the problem is fixed-parameter tractable if and only if ℓ ≤ 3. Finally, we show that our algorithmization can also be used to show that the related Connected Dominating Set problem is fixed-parameter tractable on claw-free graphs.


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  1. 1.
    Allan, R.B., Laskar, R.: On Domination and Independent Domination Numbers of a Graph. Discrete Mathematics 23, 73–76 (1978)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. Journal of Computer and System Sciences 75(8), 423–434 (2009)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chudnovsky, M., Fradkin, A.O.: An approximate version of Hadwiger’s conjecture for claw-free graphs. Journal of Graph Theory 63(4), 259–278 (2010)MathSciNetMATHGoogle Scholar
  4. 4.
    Chudnovsky, M., Ovetsky, A.: Coloring Quasi-Line Graphs. Journal of Graph Theory 54(1), 41–50 (2007)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Chudnovsky, M., Seymour, P.D.: The Structure of Claw-Free Graphs. London Mathematical Society Lecture Note Series: Surveys in Combinatorics, vol. 327, pp. 153–171 (2005)Google Scholar
  6. 6.
    Chudnovsky, M., Seymour, P.D.: Claw-free graphs. V. Global structure. Journal of Combinatorial Theory, Series B 98(6), 1373–1410 (2008)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cygan, M., Philip, G., Pilipczuk, M., Pilipczuk, M., Wojtaszczyk, J.O.: Dominating Set is Fixed Parameter Tractable in Claw-free Graphs, arXiv:1011.6239v1 (cs.DS) (2010) (preprint)Google Scholar
  8. 8.
    Damaschke, P.: Parameterized Enumeration, Transversals, and Imperfect Phylogeny Reconstruction. In: Downey, R., Fellows, M., Dehne, P. (eds.) IWPEC 2004. LNCS, vol. 3162, pp. 1–12. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  9. 9.
    Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness. Congressus Numerantium 87, 161–178 (1992)MathSciNetMATHGoogle Scholar
  10. 10.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999)CrossRefMATHGoogle Scholar
  11. 11.
    Edmonds, J.: Paths, trees, and flowers. Canadian Journal of Mathematics 17, 449–467 (1965)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Eisenbrand, F., Oriolo, G., Stauffer, G., Ventura, P.: The Stable Set Polytope of Quasi-Line Graphs. Combinatorica 28(1), 45–67 (2008)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Faudree, R., Flandrin, E., Ryjáček, Z.: Claw-free graphs – A survey. Discrete Mathematics 164(1-3), 87–147 (1997)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Faenza, Y., Oriolo, G., Stauffer, G.: An algorithmic decomposition of claw-free graphs leading to an O(n 3)-algorithm for the weighted stable set problem. In: Proc. SODA 2011, pp. 630–646. ACM-SIAM (2011)Google Scholar
  15. 15.
    Feige, U.: A Threshold of ln n for Approximating Set Cover. Journal of the ACM 45, 634–652 (1998)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Fernau, H.: Edge Dominating Set: Efficient Enumeration-Based Exact Algorithms. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 142–153. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  17. 17.
    Galluccio, A., Gentile, C., Ventura, P.: The stable set polytope of claw-free graphs with large stability number. Electronic Notes Discrete Mathematics 36, 1025–1032 (2010)CrossRefMATHGoogle Scholar
  18. 18.
    Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of domination in graphs. Marcel Dekker Inc., New York (1998)MATHGoogle Scholar
  19. 19.
    Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Domination in graphs: Advanced Topics. Marcel Dekker Inc., New York (1998)MATHGoogle Scholar
  20. 20.
    Hsu, W.-L., Tsai, K.-H.: Linear-time algorithms on circular arc graphs. Information Processing Letters 40, 123–129 (1991)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Karp, R.M.: Reducibility Among Combinatorial Problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum, New YorkGoogle Scholar
  22. 22.
    King, A.D.: Claw-free graphs and two conjectures on ω, Δ, and χ, PhD Thesis, McGill University, Montreal, Canada (2009)Google Scholar
  23. 23.
    King, A.D., Reed, B.A.: Bounding χ in Terms of ω and Δ for Quasi-Line Graphs. Journal of Graph Theory 59(3), 215–228 (2008)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Marx, D.: Parameterized Complexity of Independence and Domination on Geometric Graphs. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 154–165. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  25. 25.
    Minty, G.J.: On maximal independent sets of vertices in claw-free graphs. Journal of Combinatorial Theory, Series B 28(3), 284–304 (1980)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Misra, N., Philip, G., Raman, V., Saurabh, S.: The effect of girth on the kernelization complexity of Connected Dominating Set. In: Lodaya, K., Mahajan, M. (eds.) FSTTCS 2010, Schloss Dagstuhl, Daghstuhl, Germany. LIPIcs, vol. 8, pp. 96–107 (2010)Google Scholar
  27. 27.
    Nakamura, D., Tamura, A.: A revision of Minty’s algorithm for finding a maximum weighted stable set of a claw-free graph. Journal of the Operations Research Society of Japan 44(2), 194–204 (2001)MathSciNetMATHGoogle Scholar
  28. 28.
    Philip, G., Raman, V., Sikdar, S.: Solving Dominating Set in Larger Classes of Graphs: FPT Algorithms and Polynomial Kernels. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 694–705. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  29. 29.
    Plesník, J.: Constrained weighted matchings and edge coverings in graphs. Discrete Applied Mathematics 92(2-3), 229–241 (1999)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Sbihi, N.: Algorithme de recherche d’un stable de cardinalité maximum dans un graphe sans étoile. Discrete Mathematics 29(1), 53–76 (1980)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Stauffer, G.: Personal communicationGoogle Scholar
  32. 32.
    Yannakakis, M., Gavril, F.: Edge dominating sets in graphs. SIAM Journal on Applied Mathematics 38(3), 364–372 (1980)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Danny Hermelin
    • 1
  • Matthias Mnich
    • 2
  • Erik Jan van Leeuwen
    • 3
  • Gerhard J. Woeginger
    • 4
  1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany
  2. 2.International Computer Science InstituteBerkeleyUSA
  3. 3.Department of InformaticsUniversity of BergenNorway
  4. 4.Department of Mathematics and Computer ScienceTU EindhovenThe Netherlands

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