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Domination When the Stars Are Out

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Automata, Languages and Programming (ICALP 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6755))

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Abstract

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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References

  1. Allan, R.B., Laskar, R.: On Domination and Independent Domination Numbers of a Graph. Discrete Mathematics 23, 73–76 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  4. Chudnovsky, M., Ovetsky, A.: Coloring Quasi-Line Graphs. Journal of Graph Theory 54(1), 41–50 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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. Chudnovsky, M., Seymour, P.D.: Claw-free graphs. V. Global structure. Journal of Combinatorial Theory, Series B 98(6), 1373–1410 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  9. Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness. Congressus Numerantium 87, 161–178 (1992)

    MathSciNet  MATH  Google Scholar 

  10. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999)

    Book  MATH  Google Scholar 

  11. Edmonds, J.: Paths, trees, and flowers. Canadian Journal of Mathematics 17, 449–467 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  12. Eisenbrand, F., Oriolo, G., Stauffer, G., Ventura, P.: The Stable Set Polytope of Quasi-Line Graphs. Combinatorica 28(1), 45–67 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Faudree, R., Flandrin, E., Ryjáček, Z.: Claw-free graphs – A survey. Discrete Mathematics 164(1-3), 87–147 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  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. Feige, U.: A Threshold of ln n for Approximating Set Cover. Journal of the ACM 45, 634–652 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  18. Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of domination in graphs. Marcel Dekker Inc., New York (1998)

    MATH  Google Scholar 

  19. Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Domination in graphs: Advanced Topics. Marcel Dekker Inc., New York (1998)

    MATH  Google Scholar 

  20. Hsu, W.-L., Tsai, K.-H.: Linear-time algorithms on circular arc graphs. Information Processing Letters 40, 123–129 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  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 York

    Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  29. Plesník, J.: Constrained weighted matchings and edge coverings in graphs. Discrete Applied Mathematics 92(2-3), 229–241 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  30. Sbihi, N.: Algorithme de recherche d’un stable de cardinalité maximum dans un graphe sans étoile. Discrete Mathematics 29(1), 53–76 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  31. Stauffer, G.: Personal communication

    Google Scholar 

  32. Yannakakis, M., Gavril, F.: Edge dominating sets in graphs. SIAM Journal on Applied Mathematics 38(3), 364–372 (1980)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Max-Planck-Institut für Informatik, Saarbrücken, Germany

    Danny Hermelin

  2. International Computer Science Institute, Berkeley, USA

    Matthias Mnich

  3. Department of Informatics, University of Bergen, Norway

    Erik Jan van Leeuwen

  4. Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands

    Gerhard J. Woeginger

Authors
  1. Danny Hermelin
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  2. Matthias Mnich
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  3. Erik Jan van Leeuwen
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  4. Gerhard J. Woeginger
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Editor information

Editors and Affiliations

  1. ICE-TCS, School of Computer Science, Reykjavik University, Menntavegur 1, IS 101, Reykjavik, Iceland

    Luca Aceto

  2. Fakultät für Informatik, Universität Wien, Universitätsstraße10/9, 1090, Wien, Österreich, Austria

    Monika Henzinger

  3. Department of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00, Praha 1, Czech Republic

    Jiří Sgall

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© 2011 Springer-Verlag Berlin Heidelberg

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Hermelin, D., Mnich, M., van Leeuwen, E.J., Woeginger, G.J. (2011). Domination When the Stars Are Out. In: Aceto, L., Henzinger, M., Sgall, J. (eds) Automata, Languages and Programming. ICALP 2011. Lecture Notes in Computer Science, vol 6755. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22006-7_39

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  • DOI: https://doi.org/10.1007/978-3-642-22006-7_39

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-22005-0

  • Online ISBN: 978-3-642-22006-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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