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Blocking Independent Sets for H-Free Graphs via Edge Contractions and Vertex Deletions

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 10185)

Abstract

Let d and k be two given integers, and let G be a graph. Can we reduce the independence number of G by at least d via at most k graph operations from some fixed set S? This problem belongs to a class of so-called blocker problems. It is known to be co-NP-hard even if S consists of either an edge contraction or a vertex deletion. We further investigate its computational complexity under these two settings:

  • we give a sufficient condition on a graph class for the vertex deletion variant to be co-NP-hard even if \(d=k=1\);

  • in addition we prove that the vertex deletion variant is co-NP-hard for triangle-free graphs even if \(d=k=1\);

  • we prove that the edge contraction variant is NP-hard for bipartite graphs but linear-time solvable for trees.

By combining our new results with known ones we are able to give full complexity classifications for both variants restricted to H-free graphs.

D. Paulusma received support from EPSRC (EP/K025090/1).

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References

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

    MathSciNet  CrossRef  MATH  Google Scholar 

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

    MathSciNet  CrossRef  MATH  Google Scholar 

  3. Bentz, C., Costa, M.-C., de Werra, D., Picouleau, C., Ries, B.: Weighted transversals and blockers for some optimization problems in graphs. In: Progress in Combinatorial Optimization, Wiley-ISTE (2012)

    Google Scholar 

  4. Boros, E., Golumbic, M.C., Levit, V.E.: On the number of vertices belonging to all maximum stable sets of a graph. Discrete Appl. Math. 124, 17–25 (2002)

    MathSciNet  CrossRef  MATH  Google Scholar 

  5. Costa, M.-C., de Werra, D., Picouleau, C.: Minimum \(d\)-blockers and \(d\)-transversals in graphs. J. Comb. Optim. 22, 857–872 (2011)

    MathSciNet  CrossRef  MATH  Google Scholar 

  6. Diestel, R.: Graph Theory. Springer, Heidelberg (2005)

    MATH  Google Scholar 

  7. Diner, Ö.Y., Paulusma, D., Picouleau, C., Ries, B.: Contraction blockers for graphs with forbidden induced paths. In: Paschos, V.T., Widmayer, P. (eds.) CIAC 2015. LNCS, vol. 9079, pp. 194–207. Springer, Cham (2015). doi:10.1007/978-3-319-18173-8_14

    CrossRef  Google Scholar 

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

    Google Scholar 

  9. Golovach, P.A., Heggernes, P., Hof, P.V., Paul, C.: Hadwiger number of graphs with small chordality. In: Kratsch, D., Todinca, I. (eds.) WG 2014. LNCS, vol. 8747, pp. 201–213. Springer, Cham (2014). doi:10.1007/978-3-319-12340-0_17

    Google Scholar 

  10. Hammer, P.L., Hansen, P., Simeone, B.: Vertices belonging to all or to no maximum stable sets of a graph. SIAM J. Algebraic Discrete Methods 3, 511–522 (1982)

    MathSciNet  CrossRef  MATH  Google Scholar 

  11. Levit, V.E., Mandrescu, E.: Combinatorial properties of the family of maximum stable sets of a graph. Discrete Appl. Math. 117, 149–161 (2002)

    MathSciNet  CrossRef  MATH  Google Scholar 

  12. Levit, V.E., Mandrescu, E.: Vertices belonging to all critical sets of a graph. SIAM J. Discrete Math. 26, 399–403 (2012)

    MathSciNet  CrossRef  MATH  Google Scholar 

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

    MathSciNet  CrossRef  Google Scholar 

  14. Paulusma, D., Picouleau, C., Ries, B.: Reducing the clique and chromatic number via edge contractions and vertex deletions. In: Cerulli, R., Fujishige, S., Mahjoub, A.R. (eds.) ISCO 2016. LNCS, vol. 9849, pp. 38–49. Springer, Cham (2016). doi:10.1007/978-3-319-45587-7_4

    CrossRef  Google Scholar 

  15. Poljak, S.: A note on the stable sets and coloring of graphs. Comment. Math. Univ. Carol. 15, 307–309 (1974)

    MathSciNet  MATH  Google Scholar 

  16. 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 Math. 310, 132–146 (2010)

    MathSciNet  CrossRef  MATH  Google Scholar 

  17. Savage, C.: Maximum matchings and trees. Inf. Process. Lett. 10, 202–205 (1980)

    MathSciNet  CrossRef  MATH  Google Scholar 

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

    Google Scholar 

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Correspondence to Bernard Ries .

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Paulusma, D., Picouleau, C., Ries, B. (2017). Blocking Independent Sets for H-Free Graphs via Edge Contractions and Vertex Deletions. In: Gopal, T., Jäger , G., Steila, S. (eds) Theory and Applications of Models of Computation. TAMC 2017. Lecture Notes in Computer Science(), vol 10185. Springer, Cham. https://doi.org/10.1007/978-3-319-55911-7_34

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  • DOI: https://doi.org/10.1007/978-3-319-55911-7_34

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