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Large Independent Sets in Subquartic Planar Graphs

  • Matthias Mnich
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9627)

Abstract

By the famous Four Color Theorem, every planar graph admits an independent set that contains at least one quarter of its vertices. This lower bound is tight for infinitely many planar graphs, and finding maximum independent sets in planar graphs is \(\mathsf {NP}\)-hard. A well-known open question in the field of Parameterized Complexity asks whether the problem of finding a maximum independent set in a given planar graph is fixed-parameter tractable, for parameter the “gain” over this tight lower bound. This open problem has been posed many times [4, 8, 10, 13, 17, 20, 31, 32, 35, 38].

We show fixed-parameter tractability of the independent set problem parameterized above tight lower bound in planar graphs with maximum degree at most 4, in subexponential time.

Keywords

Planar Graph Tree Decomposition Reduction Rule Path Decomposition Color Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

I am indebted to Zdeněk Dvořák for helpful remarks, and an anonymous reviewer who suggested considering treewidth over pathwidth.

References

  1. 1.
    Albertson, M., Bollobas, B., Tucker, S.: The independence ratio and maximum degree of a graph. In: Proceedings of the Seventh Southeastern Conference on Combinatorics, Graph Theory and Computing, pp. 43–50. Congressus Numerantium, No. XVII. Utilitas Math., Winnipeg, Man. (1976)Google Scholar
  2. 2.
    Appel, K., Haken, W.: Every planar map is four colorable. Bull. Amer. Math. Soc. 82(5), 711–712 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berge, C.: Graphs and Hypergraphs, revised edn. North-Holland Publishing Co., Amsterdam (1976)zbMATHGoogle Scholar
  4. 4.
    Bodlaender, H.L.: Open problems in parameterized and exact computation. Technical report UU-CS-2008-017, Utrecht University (2008)Google Scholar
  5. 5.
    Brooks, R.L.: On colouring the nodes of a network. Proc. Camb. Philos. Soc. 37, 194–197 (1941)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cai, L., Juedes, D.: On the existence of subexponential parameterized algorithms. J. Comput. Syst. Sci. 67(4), 789–807 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cranston, D.W., Rabern, L.: Planar graphs are 9/2-colorable and have independence ratio at least 3/13 (2015). http://arxiv.org/abs/1410.7233
  8. 8.
    Crowston, R., Fellows, M., Gutin, G., Jones, M., Rosamond, F., Thomassé, S., Yeo, A.: Simultaneously satisfying linear equations over \(\mathbb{F}_2\): MaxLin2 and Max-\(r\)-Lin2 parameterized above average. In: Proceedings of FSTTCS 2011, pp. 229–240 (2011)Google Scholar
  9. 9.
    Crowston, R., Jones, M., Mnich, M.: Max-cut parameterized above the Edwards-Erdős bound. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part I. LNCS, vol. 7391, pp. 242–253. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  10. 10.
    Cygan, M., Fomin, F., Jansen, B., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Open problems from the Bedlewo school on parameterized algorithms and complexity (2014). http://fptschool.mimuw.edu.pl/opl.pdf
  11. 11.
    Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, New York (2015)CrossRefzbMATHGoogle Scholar
  12. 12.
    Cygan, M., Pilipczuk, M., Pilipczuk, M., Wojtaszczyk, J.O.: On multiway cut parameterized above lower bounds. ACM Trans. Comput. Theory 5(1), 3:1–3:11 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dvořák, Z., Mnich, M.: Large independent sets in triangle-free planar graphs. In: Schulz, A.S., Wagner, D. (eds.) ESA 2014. LNCS, vol. 8737, pp. 346–357. Springer, Heidelberg (2014)Google Scholar
  14. 14.
    Dvořák, Z., Sereni, J.-S.S., Volec, J.: Subcubic triangle-free graphs have fractional chromatic number at most 14/5. J. London Math. Soc. 89(3), 641–662 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Eppstein, D.: Subgraph isomorphism in planar graphs and related problems. J. Graph Algorithms Appl. 3(3), 27 (electronic) (1999)Google Scholar
  16. 16.
    Faria, L., Klein, S., Stehlík, M.: Odd cycle transversals and independent sets in fullerene graphs. SIAM J. Discrete Math. 26(3), 1458–1469 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fellows, M.R., Guo, J., Marx, D., Saurabh, S.: Data reduction and problem kernels (Dagstuhl Seminar 12241). Dagstuhl Reports 2(6), 26–50 (2012)Google Scholar
  18. 18.
    Fleischner, H., Sabidussi, G., Sarvanov, V.I.: Maximum independent sets in 3- and 4-regular Hamiltonian graphs. Discrete Math. 310(20), 2742–2749 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. Freeman, New York (1979)zbMATHGoogle Scholar
  20. 20.
    Giannopoulou, A.C., Kolay, S., Saurabh, S.: New lower bound on Max Cut of hypergraphs with an application to r-Set Splitting. In: Fernández-Baca, D. (ed.) LATIN 2012. LNCS, vol. 7256, pp. 408–419. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  21. 21.
    Grohe, M.: Local tree-width, excluded minors, and approximation algorithms. Combinatorica 23(4), 613–632 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Grötzsch, H.: Zur Theorie der diskreten Gebilde. VII. Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel. Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg. Math.-Nat. Reihe, 8, 109–120 (1958/1959)Google Scholar
  23. 23.
    Gutin, G., Jones, M., Yeo, A.: Kernels for below-upper-bound parameterizations of the hitting set and directed dominating set problems. Theoret. Comput. Sci. 412(41), 5744–5751 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Gutin, G., Kim, E.J., Mnich, M., Yeo, A.: Betweenness parameterized above tight lower bound. J. Comput. System Sci. 76(8), 872–878 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Gutin, G., van Iersel, L., Mnich, M., Yeo, A.: Every ternary permutation constraint satisfaction problem parameterized above average has a kernel with a quadratic number of variables. J. Comput. System Sci. 78(1), 151–163 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Heckman, C.C., Thomas, R.: Independent sets in triangle-free cubic planar graphs. J. Combin. Theory Ser. B 96(2), 253–275 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kammer, F., Tholey, T.: Approximate tree decompositions of planar graphs in linear time. In: Proceedings of SODA 2012, pp. 683–698 (2012)Google Scholar
  28. 28.
    King, A.D., Lu, L., Peng, X.: A fractional analogue of Brooks’ theorem. SIAM J. Discrete Math. 26(2), 452–471 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Lu, L., Peng, X.: The fractional chromatic number of triangle-free graphs with \(\Delta \le 3\). Discrete Math. 312(24), 3502–3516 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Mahajan, M., Raman, V.: Parameterizing above guaranteed values: MaxSat and MaxCut. J. Algorithms 31(2), 335–354 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Mahajan, M., Raman, V., Sikdar, S.: Parameterizing above or below guaranteed values. J. Comput. System Sci. 75(2), 137–153 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Mnich, M.: Algorithms in moderately exponential time. Ph.D. thesis, TU Eindhoven (2010)Google Scholar
  33. 33.
    Mnich, M., Zenklusen, R.: Bisections above tight lower bounds. In: Golumbic, M.C., Stern, M., Levy, A., Morgenstern, G. (eds.) WG 2012. LNCS, vol. 7551, pp. 184–193. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  34. 34.
    Molloy, M., Reed, B.: Graph Colouring and the Probabilistic Method. Algorithms and Combinatorics, vol. 23. Springer, Berlin (2002)zbMATHGoogle Scholar
  35. 35.
    Niedermeier, R.: Invitation to Fixed-parameter Algorithms. Oxford Lecture Series in Mathematics and its Applications, vol. 31. OUP, Oxford (2006)CrossRefzbMATHGoogle Scholar
  36. 36.
    Robertson, N., Sanders, D., Seymour, P., Thomas, R.: The four-colour theorem. J. Combin. Theory Ser. B 70(1), 2–44 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Scheinerman, E.R., Ullman, D.H.: Fractional Graph Theory. Dover Publications Inc., Mineola (2011)zbMATHGoogle Scholar
  38. 38.
    Sikdar, S.: Parameterizing from the extremes. Ph.D. thesis, The Institute of Mathematical Sciences, Chennai (2010)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Universität BonnBonnGermany

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