Graphs and Combinatorics

, Volume 31, Issue 1, pp 73–90 | Cite as

Blockers for the Stability Number and the Chromatic Number

  • C. Bazgan
  • C. Bentz
  • C. PicouleauEmail author
  • B. Ries
Original Paper


Given an undirected graph G = (V, E) and two positive integers k and d, we are interested in finding a set of edges (resp. non-edges) of size at most k to delete (resp. to add) in such a way that the chromatic number (resp. stability number) in the resulting graph will decrease by at least d compared to the original graph. We investigate these two problems in various classes of graphs (split graphs, threshold graphs, bipartite graphs and their complements) and determine their computational complexity. In some of the polynomial-time solvable cases, we also give a structural description of a solution.


Blocker Chromatic number Stability number Bipartitegraph Split graph Threshold graph 


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  1. 1.
    Asratian, A., Denley, T.M.J., Häggkvist, R.: Bipartite Graphs and Their Applications. Cambridge University Press, Cambridge (1998)Google Scholar
  2. 2.
    Bar-Noy, A., Khuller, S., Schieber, B.: The complexity of finding most vital arcs and nodes, Technical Report CS-TR-3539, University of Maryland (1995)Google Scholar
  3. 3.
    Bazgan C., Toubaline S., Tuza Z.: The most vital nodes with respect to independent set and vertex cover. Discrete Appl. Math. 159(17), 1933–1946 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bazgan, C., Toubaline, S., Tuza, Z.: Complexity of the most vital nodes for independent set on tree structures. In: Proceedings of the 21 International Workshop on Combinatorial Algorithms (IWOCA 2010), LNCS vol. 6460, pp. 154–166 (2010)Google Scholar
  5. 5.
    Bazgan C., Toubaline S., Vanderpooten D.: Complexity of determining the most vital elements for the p-median and p-center location problems. J. Comb. Optim. 25(2), 191–207 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Bazgan C., Toubaline S., Vanderpooten D.: Critical edges/nodes for the minimum spanning tree problem: complexity and approximation. J. Comb. Optim. 26, 178–189 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Bentz C., Costa M.-C., Picouleau C., Ries B., de Werra D.: d-Transversals of stable sets and vertex covers in weighted bipartite graphs. J. Discrete Algorithms 17, 95–102 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Brooks R.L.: On colouring the nodes of a network. Proc. Camb. Philos. Soc. Math. Phys. Sci. 37, 194–197 (1941)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Chlebík, M., Chlebíková, J.: Crown reductions for the minimum weighted vertex cover problem. Discrete Appl. Math. 156(3), 292–312 (2008)Google Scholar
  10. 10.
    Costa M.-C., de Werra D., Picouleau C.: Minimum d-blockers and d-transversals in graphs. J. Comb. Optim. 22(4), 857–872 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Fink J.F., Jacobson M.S., Kinch L.F., Roberts J.: The bondage number of a graph. Discrete Math. 86(1-3), 47–57 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Frederickson, G.N., Solis-Oba, R.: Increasing the weight of minimum spanning trees. In: Proceedings of the 7 ACM-SIAM Symposium on Discrete Algorithms (SODA 1996) pp. 539–546Google Scholar
  13. 13.
    Garey M.R., Johnson D.S.: Computers and tractability: a guide to the theory of NP-completeness. Freeman, New York (1979)Google Scholar
  14. 14.
    Khachiyan L., Boros E., Borys K., Elbassioni K., Gurvich V., Rudolf G., Zhao J.: On short paths interdiction problems: total and node-wise limited interdiction. Theory Comput. Syst. 43(2), 204–233 (2008)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Monnot J., Toulouse S.: The path partition problem and related problems in bipartite graphs. Oper. Res. Lett. 35(5), 677–684 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 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(1), 132–146 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Toubaline, S.: (2010) Détermination des éléments les plus vitaux pour des problèmes de graphes, PhD Thesis, Université Paris-DauphineGoogle Scholar
  18. 18.
    Turán P.: On an extremal problem in graph theory (in Hungarian). Matematikai s Fizikai Lapok. 48, 436–452 (1941)Google Scholar
  19. 19.
    West, D.B.: (2001) Introduction to Graph Theory, 2nd Edn. Prentice Hall, Englewood CliffsGoogle Scholar
  20. 20.
    Wood R.K.: Deterministic network interdiction. Math. Comput. Model. 17(2), 1–18 (1993)CrossRefzbMATHGoogle Scholar
  21. 21.
    Zenklusen, R.: Matching Interdiction. Discrete Appl. Math. 158, 1676–1690 (2010)Google Scholar
  22. 22.
    Zenklusen R., Ries B., Picouleau C., de Werra D., Costa M.-C., Bentz C.: Blockers and transversals. Discrete Math. 309(13), 4306–4314 (2009)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Japan 2013

Authors and Affiliations

  1. 1.PSL, LAMSADE, CNRS UMR 7243Université Paris-DauphineParisFrance
  2. 2.Institut Universitaire de FranceParisFrance
  3. 3.CEDRIC-CNAMParisFrance

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