Advertisement

Maximum Weight Independent Sets in (\(S_{1,1,3}\), bull)-free Graphs

  • T. KarthickEmail author
  • Frédéric Maffray
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9797)

Abstract

The Maximum Weight Independent Set (MWIS) problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. The MWIS problem is well known to be NP-complete in general, even under substantial restrictions. The computational complexity of the MWIS problem for \(S_{1, 1, 3}\)-free graphs is unknown. In this note, we give a proof for the solvability of the MWIS problem for (\(S_{1, 1, 3}\), bull)-free graphs in polynomial time. Here, an \(S_{1, 1, 3}\) is the graph with vertices \(v_1, v_2, v_3, v_4, v_5, v_6\) and edges \(v_1v_2, v_2v_3, v_3v_4, v_4v_5, v_4v_6\), and the bull is the graph with vertices \(v_1, v_2, v_3, v_4, v_5\) and edges \(v_1v_2, \) \( v_2v_3, v_3v_4, \) \( v_2v_5, v_3v_5\).

Keywords

Graph algorithms Weighted independent set Modular decomposition Claw-free graph Fork-free graph Bull-free graph 

References

  1. 1.
    Alekseev, V.E.: The effect of local constraints on the complexity of determination of the graph independence number. In: Combinatorial-Algebraic Methods in Applied Mathematics, pp. 3–13. Gorkiy University Press, Gorky (1982). (in Russian)Google Scholar
  2. 2.
    Alekseev, V.E., Lozin, V.V., Malyshev, D., Milanič, M.: The maximum independent set problem in planar graphs. In: Ochmański, E., Tyszkiewicz, J. (eds.) MFCS 2008. LNCS, vol. 5162, pp. 96–107. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  3. 3.
    Arora, S., Barak, B.: Computational Complexity - A Modern Approach. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
  4. 4.
    Basavaraju, M., Chandran, L.S., Karthick, T.: Maximum weight independent sets in hole- and dart-free graphs. Discrete Appl. Math. 160, 2364–2369 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brandstädt, A., Giakoumakis, V., Maffray, F.: Clique separator decomposition of hole- and Diamond-free graphs and algorithmic consequences. Discrete Appl. Math. 160, 471–478 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brandstädt, A., Hoáng, C.T.: On clique separators, nearly chordal graphs, and the maximum weight stable set problem. Theor. Comput. Sci. 389, 295–306 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brandstädt, A., Karthick, T.: Weighted efficient domination in two subclasses of \(P_6\)-free graphs. Discrete Appl. Math. 201, 38–46 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM Monographs on Discrete Mathematics, vol. 3. SIAM, Philadelphia (1999)CrossRefzbMATHGoogle Scholar
  9. 9.
    Brandstädt, A., Lozin, V.V., Mosca, R.: Independent sets of maximum weight in apple-free graphs. SIAM J. Discrete Math. 24(1), 239–254 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Brandstädt, A., Mosca, R.: Maximum weight independent sets in odd-hole-free graphs without dart or without bull. Graphs Comb. 31, 1249–1262 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Brause, C., Le, N.C., Schiermeyer, I.: The maximum independent det problem in subclasses of subcubic graphs. Discrete Math. 338, 1766–1778 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chudnovsky, M.: The structure of bull-free graphs I: three-edge paths with centers and anticenters. J. Comb. Theor. B 102, 233–251 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chudnovsky, M.: The structure of bull-free graphs II and III: a summary. J. Comb. Theor. B 102, 252–282 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Corneil, D.G.: The complexity of generalized clique packing. Discrete Appl. Math. 12, 233–240 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Corneil, D.G., Perl, Y., Stewart, L.K.: A linear recognition for cographs. SIAM J. Comput. 14, 926–934 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999)CrossRefzbMATHGoogle Scholar
  17. 17.
    Farber, M.: On diameters and radii of bridged graphs. Discrete Math. 73, 249–260 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gerber, M.U., Hertz, A., Lozin, V.V.: Stable sets in two subclasses of banner-free graphs. Discrete Appl. Math. 132, 121–136 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1, 169–197 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Frank, A.: Some polynomial algorithms for certain graphs and hypergraphs. In: Proceedings of the Fifth BCC, Congressus Numerantium, XV, pp. 211–226 (1976)Google Scholar
  21. 21.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations, pp. 85–103. Springer, New York (1972)CrossRefGoogle Scholar
  22. 22.
    Karthick, T.: Weighted independent sets in a subclass of \(P_6\)-free graphs. Discrete Math. 339, 1412–1418 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Karthick, T., Maffray, F.: Maximum weight independent sets inclasses related to claw-free graphs. Discrete Appl. Math. (2015). http://dx.doi.org/10.1016/j.dam.2015.02.012
  24. 24.
    Le, N.C., Brause, C., Schiermeyer, I.: New sufficient conditions for \(\alpha \)-redundant vertices. Discrete Math. 338, 1674–1680 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lokshtanov, D., Vatshelle, M., Villanger, Y.: Independent set in \(P_5\)-free graphs in polynomial time. In: Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 570–581 (2014)Google Scholar
  26. 26.
    Lozin, V.V., Milanič, M.: A polynomial algorithm to find an independent set of maximum weight in a fork-free graph. J. Discrete Algorithms 6, 595–604 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Lozin, V.V., Milanič, M., Purcell, C.: Graphs without large apples and the maximum weight independent set problem. Graphs Comb. 30, 395–410 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Lozin, V.V., Monnot, J., Ries, B.: On the maximum independent set problem in subclasses of subcubic graphs. J. Discrete Algorithms 31, 104–112 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Maffray, F., Pastor, L.: The maximum weight stable set problem in (\(P_6\), bull)-free graphs (2016). arXiv:1602.06817v1
  30. 30.
    Malyshev, D.S.: Classes of subcubic planar graphs for which the indepedent set problem is polynomial-time solvable. J. Appl. Ind. Math. 7, 537–548 (2013)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Minty, G.M.: On maximal independent sets of vertices in claw-free graphs. J. Comb. Theor. Ser. B 28, 284–304 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Poljak, S.: A note on stable sets and colorings of graphs. Commun. Math. Univ. Carolinae 15, 307–309 (1974)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Reed, B., Sbihi, N.: Recognizing bull-free perfect graphs. Graphs Comb. 11, 171–178 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Thomassé, S., Trotignon, N., Vušković, K.: A polynomial Turing-kernel for weighted independent set in bull-free graphs. Algorithmica (2015, in press)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Indian Statistical InstituteChennaiIndia
  2. 2.CNRS, Laboratoire G-SCOPUniversity of Grenoble-AlpesGrenobleFrance

Personalised recommendations