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Contraction Blockers for Graphs with Forbidden Induced Paths

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 9079)

Abstract

We consider the following problem: can a certain graph parameter of some given graph be reduced by at least \(d\) for some integer \(d\) via at most \(k\) edge contractions for some given integer \(k\)? We examine three graph parameters: the chromatic number, clique number and independence number. For each of these graph parameters we show that, when \(d\) is part of the input, this problem is polynomial-time solvable on \(P_4\)-free graphs and NP-complete as well as W[1]-hard, with parameter \(d\), for split graphs. As split graphs form a subclass of \(P_5\)-free graphs, both results together give a complete complexity classification for \(P_\ell \)-free graphs. The W[1]-hardness result implies that it is unlikely that the problem is fixed-parameter tractable for split graphs with parameter \(d\). But we do show, on the positive side, that the problem is polynomial-time solvable, for each parameter, on split graphs if \(d\) is fixed, i.e., not part of the input. We also initiate a study into other subclasses of perfect graphs, namely cobipartite graphs and interval graphs.

Keywords

  • Chromatic Number
  • Interval Graph
  • Chordal Graph
  • Graph Class
  • Perfect Graph

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.

Öznur Yaşar Diner—Supported partially by Marie Curie International Reintegration Grant PIRG07/GA/2010/268322.

Daniël Paulusma—Supported by EPSRC EP/K025090/1.

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References

  1. Alekseev, V.E.: On easy and hard hereditary classes of graphs with respect to the independent set problem. Discrete Applied Math. 132, 17–26 (2004)

    CrossRef  Google Scholar 

  2. Bazgan, C., Bentz, C., Picouleau, C., Ries, B.: Blockers for the stability number and the chromatic number. Graphs and Combinatorics 31, 73–90 (2015)

    CrossRef  MATH  MathSciNet  Google Scholar 

  3. Bazgan, C., Toubaline, S., Tuza, Z.: Complexity of Most Vital Nodes for Independent Set in Graphs Related to Tree Structures. In: Iliopoulos, C.S., Smyth, W.F. (eds.) IWOCA 2010. LNCS, vol. 6460, pp. 154–166. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  4. Bazgan, C., Toubaline, S., Tuza, Z.: The most vital nodes with respect to independent set and vertex cover. Discrete Applied Mathematics 159(17), 1933–1946 (2011)

    CrossRef  MATH  MathSciNet  Google Scholar 

  5. Belmonte, R., Golovach, P.A., van’ t Hof, P.: Parameterized complexity of three edge contraction problems with degree constraints. Acta Informatica 51, 473–497 (2014)

    CrossRef  MATH  MathSciNet  Google Scholar 

  6. Bodlaender, H.L., Möhring, R.H.: The pathwidth and treewidth of cographs. SIAM J. Discrete Math. 6, 181–188 (1993)

    CrossRef  MATH  MathSciNet  Google Scholar 

  7. Brandstädt, A., Le, V.B., Spinrad, J.: Graph Classes: A Survey. SIAM Monographs on Discrete Mathematics and Applications (1999)

    Google Scholar 

  8. Chvátal, V., Hoàng, C.T., Mahadev, N.V.R., de Werra, D.: Four classes of perfectly orderable graphs. J. Graph Theory 11, 481–495 (1987)

    CrossRef  MATH  MathSciNet  Google Scholar 

  9. Corneil, D.G., Lerchs, H., Stewart Burlingham, L.: Complement reducible graphs. Discrete Applied Mathematics 3, 163–174 (1981)

    CrossRef  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)

    CrossRef  MATH  MathSciNet  Google Scholar 

  11. Costa, M.-C., de Werra, D., Picouleau, C.: Minimum d-blockers and d-transversals in graphs. Journal of Combinatorial Optimization 22, 857–872 (2011)

    CrossRef  MATH  MathSciNet  Google Scholar 

  12. Földes, S., Hammer, P.L.: Split graphs. In: 8th South-Eastern Conf. on Combinatorics, Graph Theory and Computing, Congressus Numerantium. vol. 19, pp. 311–315 (1977)

    Google Scholar 

  13. Fulkerson, D., Gross, O.: Incidence matrices and interval graphs. Pacific Journal of Mathematics 15, 835–855 (1965)

    CrossRef  MATH  MathSciNet  Google Scholar 

  14. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman (1979)

    Google Scholar 

  15. Golovach, P.A., Heggernes, P., van ’t Hof, P., Paul, C.: Hadwiger number of graphs with small chordality. In: Kratsch, D., Todinca, I. (eds.) WG 2014. LNCS, vol. 8747, pp. 201–213. Springer, Heidelberg (2014)

    CrossRef  Google Scholar 

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

    MATH  Google Scholar 

  17. Gutin, G., Jones, M., Yeo, A.: Kernels for Below-Upper-Bound Parameterizations of the Hitting Set and Directed Dominating Set Problems. Theor. Comput. Sci. 412, 5744–5751 (2011)

    CrossRef  MATH  MathSciNet  Google Scholar 

  18. Heggernes, P., van t Hof, P., Lokshtanov, D., Paul, C.: Obtaining a Bipartite Graph by Contracting Few Edges. SIAM Journal on Discrete Mathematics 27, 2143–2156 (2013)

    CrossRef  MATH  MathSciNet  Google Scholar 

  19. Král’, D., Kratochvíl, J., Tuza, Z., Woeginger, G.J.: Complexity of coloring graphs without forbidden induced subgraphs. In: Brandstädt, A., Le, V.B. (eds.) WG 2001. LNCS, vol. 2204, p. 254. Springer, Heidelberg (2001)

    CrossRef  Google Scholar 

  20. Lokshtanov, D., Vatshelle, M., Villanger, Y.: Independent Set in P5-Free Graphs in Polynomial Time. In: Proc. SODA, pp. 570–581 (2014)

    Google Scholar 

  21. Pajouh, F.M., Boginski, V., Pasiliao, E.L.: Minimum vertex blocker clique problem. Networks 64, 48–64 (2014)

    CrossRef  MathSciNet  Google Scholar 

  22. Ries, B., Bentz, C., Picouleau, C., de Werra, D., Costa, M.-C., Zenklusen, R.: Blockers and Transversals in some subclasses of bipartite graphs : when caterpillars are dancing on a grid. Discrete Mathematics 310, 132–146 (2010)

    CrossRef  MATH  MathSciNet  Google Scholar 

  23. Toubaline, S.: Détermination des éléments les plus vitaux pour des problèmes de graphes, Ph. D. thesis, Université Paris-Dauphine (2010)

    Google Scholar 

  24. Watanabe, T., Tadashi, A.E., Nakamura, A.: On the removal of forbidden graphs by edge-deletion or by edge-contraction. Discrete Applied Mathematics 3, 151–153 (1981)

    CrossRef  MATH  MathSciNet  Google Scholar 

  25. Watanabe, T., Tadashi, A.E., Nakamura, A.: On the NP-hardness of edge-deletion and -contraction problems. Discrete Applied Mathematics 6, 63–78 (1983)

    CrossRef  MATH  MathSciNet  Google Scholar 

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Diner, Ö.Y., Paulusma, D., Picouleau, C., Ries, B. (2015). Contraction Blockers for Graphs with Forbidden Induced Paths. In: Paschos, V., Widmayer, P. (eds) Algorithms and Complexity. CIAC 2015. Lecture Notes in Computer Science(), vol 9079. Springer, Cham. https://doi.org/10.1007/978-3-319-18173-8_14

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  • DOI: https://doi.org/10.1007/978-3-319-18173-8_14

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