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On Independent Vertex Sets in Subclasses of Apple-Free Graphs

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Abstract

In a finite undirected graph, an apple consists of a chordless cycle of length at least 4, and an additional vertex which is not in the cycle and sees exactly one of the cycle vertices. A graph is apple-free if it contains no induced subgraph isomorphic to an apple. Apple-free graphs are a common generalization of chordal graphs, claw-free graphs and cographs and occur in various papers. The Maximum Weight Independent Set (MWS) problem is efficiently solvable on chordal graphs, on cographs as well as on claw-free graphs. In this paper, we obtain partial results on some subclasses of apple-free graphs where our results show that the MWS problem is solvable in polynomial time. The main tool is a combination of clique separators with modular decomposition.

Our algorithms are robust in the sense that there is no need to recognize whether the input graph is in the given graph class; the algorithm either solves the MWS problem correctly or detects that the input graph is not in the given class.

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References

  1. Alekseev, V.E., Lozin, V.V.: Augmenting graphs for independent sets. Discrete Appl. Math. 145, 3–10 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  2. Brandstädt, A., Dragan, F.F.: On the linear and circular structure of (claw,net)-free graphs. Discrete Appl. Math. 129, 285–303 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. Brandstädt, A., Hoàng, C.T.: On clique separators, nearly chordal graphs and the Maximum Weight Stable Set problem. In: Jünger, M., Kaibel, V. (eds.) IPCO 2005, LNCS 3509, pp. 265–275 (2005)

  4. Brandstädt, A., Lozin, V.V.: A note on α-redundant vertices in graphs. Discrete Appl. Math. 108, 301–308 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  5. Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM Monographs on Discrete Math. Appl., vol. 3. SIAM, Philadelphia (1999)

    MATH  Google Scholar 

  6. Brandstädt, A., Dragan, F.F., Köhler, E.: Linear time algorithms for Hamiltonian problems on (claw,net)-free graphs. SIAM J. Comput. 30, 1662–1677 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  7. Brandstädt, A., Le, V.B., Mahfud, S.: New applications of clique separator decomposition for the Maximum Weight Stable Set problem. Theor. Comp. Sci. 370, 229–239 (2007). Extended abstract in: FCT 2005, LNCS 3623, 505–516 (2005)

    Article  MATH  Google Scholar 

  8. Chudnovsky, M., Seymour, P.: The structure of clawfree graphs. In: Surveys in Combinatorics 2005. London Math. Soc. Lecture Note Series, vol. 327, pp. 153–171. Cambridge University Press, Cambridge (2005)

    Google Scholar 

  9. Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem. Ann. Math. 164, 51–229 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Corneil, D.G., Perl, Y., Stewart, L.K.: A linear recognition algorithm for cographs. SIAM J. Comput. 14, 926–934 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  11. De Simone, C.: On the vertex packing problem. Graphs Comb. 9, 19–30 (1993)

    Article  MATH  Google Scholar 

  12. Duffus, D., Gould, R.J., Jacobson, M.S.: Forbidden subgraphs and the Hamiltonian theme. In: The Theory and Applications of Graphs (Kalamazoo, Mich. 1980), pp. 297–316. Wiley, New York (1981)

    Google Scholar 

  13. Frank, A.: Some polynomial algorithms for certain graphs and hypergraphs. In: Proceedings of the Fifth British Combinatorial Conference, pp. 211–226. Univ. Aberdeen, Aberdeen (1975). Congressus Numerantium No. XV. Utilitas Math., Winnipeg (1976)

  14. Gerber, M.U., Lozin, V.V.: Robust algorithms for the stable set problem. Graphs Comb. 19, 347–356 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gerber, M.U., Hertz, A., Lozin, V.V.: Stable sets in two subclasses of banner-free graphs. Discrete Appl. Math. 132, 121–136 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, San Diego (1980)

    MATH  Google Scholar 

  17. Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1, 169–197 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hammer, P.L., Mahadev, N.V.R., de Werra, D.: The struction of a graph: Application to CN-free graphs. Combinatorica 5, 141–147 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kelmans, A.: On Hamiltonicity of (claw,net)-free graphs. Discrete Math. 306, 2755–2761 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Lozin, V.V.: Stability in P 5- and banner-free graphs. Eur. J. Oper. Res. 125, 292–297 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  21. Lozin, V.V., Milanič, M.: Maximum independent sets in graphs of low degree. In: Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms SODA, pp. 874–880 (2007)

  22. Mahadev, N.V.R.: Vertex deletion and stability number. Research Report OR WR 90/2, Dept. of Mathematics, Swiss Federal Institute of Technology (1990)

  23. McConnell, R.M., Spinrad, J.: Modular decomposition and transitive orientation. Discrete Math. 201, 189–241 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  24. Milanič, M.: Algorithmic Developments and Complexity Results for Finding Maximum and Exact Independent Sets in Graphs. Rutgers University Press, New Brunswick (2007)

    Google Scholar 

  25. Minty, G.M.: On maximal independent sets of vertices in claw-free graphs. J. Comb. Theory B 28, 284–304 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  26. Möhring, R.H., Radermacher, F.J.: Substitution decomposition for discrete structures and connections with combinatorial optimization. Ann. Discrete Math. 19, 257–356 (1984)

    Google Scholar 

  27. Mosca, R.: Stable sets in certain P 6-free graphs. Discrete Appl. Math. 92, 177–191 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  28. Mosca, R.: Stable sets of maximum weight in (P 7,banner)-free graphs. Discrete Math. (2008, to appear)

  29. Nakamura, D., Tamura, A.: A revision of Minty’s algorithm for finding a maximum weighted stable set of a claw-free graph. J. Oper. Res. Soc. Jpn. 44, 194–204 (2001)

    MATH  MathSciNet  Google Scholar 

  30. Olariu, S.: The strong perfect graph conjecture for pan-free graphs. J. Comb. Theory B 47, 187–191 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  31. Poljak, S.: A note on stable sets and colorings of graphs. Commun. Math. Univ. Carolinae 15, 307–309 (1974)

    MATH  MathSciNet  Google Scholar 

  32. Spinrad, J.P.: Efficient Graph Representations. Fields Institute Monographs, vol. 19. American Mathematical Society, Providence (2003)

    MATH  Google Scholar 

  33. Tarjan, R.E.: Decomposition by clique separators. Discrete Math. 55, 221–232 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  34. Whitesides, S.H.: A method for solving certain graph recognition and optimization problems, with applications to perfect graphs. In: Berge, C., Chvátal, V. (eds.) Topics on Perfect Graphs. North-Holland, Amsterdam (1984)

    Google Scholar 

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Authors and Affiliations

  1. Institut für Informatik, Universität Rostock, 18051, Rostock, Germany

    Andreas Brandstädt & Tilo Klembt

  2. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

    Vadim V. Lozin

  3. Dipartimento di Scienze, Universitá degli Studi “G. D’Annunzio”, Pescara, 65121, Italy

    Raffaele Mosca

Authors
  1. Andreas Brandstädt
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  2. Tilo Klembt
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  3. Vadim V. Lozin
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  4. Raffaele Mosca
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Correspondence to Tilo Klembt.

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Brandstädt, A., Klembt, T., Lozin, V.V. et al. On Independent Vertex Sets in Subclasses of Apple-Free Graphs. Algorithmica 56, 383–393 (2010). https://doi.org/10.1007/s00453-008-9176-0

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