On the Tractability of Covering a Graph with 2-Clubs

  • Riccardo DondiEmail author
  • Manuel Lafond
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11651)


Covering a graph with cohesive subgraphs is a classical problem in theoretical computer science. In this paper, we prove new complexity results on the Open image in new window problem, a variant recently introduced in the literature which asks to cover the vertices of a graph with a minimum number of 2-clubs (which are induced subgraphs of diameter at most 2). First, we answer an open question on the decision version of Open image in new window that asks if it is possible to cover a graph with at most two 2-clubs, and we prove that it is W[1]-hard when parameterized by the distance to a 2-club. Then, we consider the complexity of Open image in new window on some graph classes. We prove that Open image in new window remains NP-hard on subcubic planar graphs, W[2]-hard on bipartite graphs when parameterized by the number of 2-clubs in a solution and fixed-parameter tractable on graphs having bounded treewidth.


  1. 1.
    Alba, R.D.: A graph-theoretic definition of a sociometric clique. J. Math. Sociol. 3, 113–126 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Asahiro, Y., Doi, Y., Miyano, E., Samizo, K., Shimizu, H.: Optimal approximation algorithms for maximum distance-bounded subgraph problems. Algorithmica 80(6), 1834–1856 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Balasundaram, B., Butenko, S., Trukhanov, S.: Novel approaches for analyzing biological networks. J. Comb. Optim. 10(1), 23–39 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bourjolly, J., Laporte, G., Pesant, G.: An exact algorithm for the maximum k-club problem in an undirected graph. Eur. J. Oper. Res. 138(1), 21–28 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Cerioli, M.R., Faria, L., Ferreira, T.O., Martinhon, C.A.J., Protti, F., Reed, B.A.: Partition into cliques for cubic graphs: planar case, complexity and approximation. Discret. Appl. Math. 156(12), 2270–2278 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Cerioli, M.R., Faria, L., Ferreira, T.O., Protti, F.: A note on maximum independent sets and minimum clique partitions in unit disk graphs and penny graphs: complexity and approximation. RAIRO-Theor. Inform. Appl. 45(3), 331–346 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Chang, M., Hung, L., Lin, C., Su, P.: Finding large k-clubs in undirected graphs. Computing 95(9), 739–758 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Dondi, R., Mauri, G., Sikora, F., Zoppis, I.: Covering a graph with clubs. J. Graph Algorithms Appl. 23(2), 271–292 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Dondi, R., Mauri, G., Zoppis, I.: On the tractability of finding disjoint clubs in a network. Theor. Comput. Sci. (2019, to appear)Google Scholar
  10. 10.
    Dumitrescu, A., Pach, J.: Minimum clique partition in unit disk graphs. Graphs Comb. 27(3), 399–411 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Fellows, M.R., Hermelin, D., Rosamond, F.A., Vialette, S.: On the parameterized complexity of multiple-interval graph problems. Theor. Comput. Sci. 410(1), 53–61 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Garey, M.R., Johnson, D.S., Stockmeyer, L.J.: Some simplified NP-complete graph problems. Theor. Comput. Sci. 1(3), 237–267 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)zbMATHGoogle Scholar
  14. 14.
    Golovach, P.A., Heggernes, P., Kratsch, D., Rafiey, A.: Finding clubs in graph classes. Discrete Appl. Math. 174, 57–65 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Hartung, S., Komusiewicz, C., Nichterlein, A.: Parameterized algorithmics and computational experiments for finding 2-clubs. J. Graph Algorithms Appl. 19(1), 155–190 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Proceedings of a symposium on the Complexity of Computer Computations, held 20–22 March 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York. The IBM Research Symposia Series, pp. 85–103. Plenum Press, New York (1972)Google Scholar
  17. 17.
    Kloks, T.: Treewidth, Computations and Approximations. LNCS, vol. 842. Springer, Heidelberg (1994). Scholar
  18. 18.
    Komusiewicz, C.: Multivariate algorithmics for finding cohesive subnetworks. Algorithms 9(1), 21 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Komusiewicz, C., Sorge, M.: An algorithmic framework for fixed-cardinality optimization in sparse graphs applied to dense subgraph problems. Discrete Appl. Math. 193, 145–161 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Laan, S., Marx, M., Mokken, R.J.: Close communities in social networks: boroughs and 2-clubs. Soc. Netw. Anal. Min. 6(1), 20:1–20:16 (2016)CrossRefGoogle Scholar
  21. 21.
    Mokken, R.: Cliques, clubs and clans. Qual. Quant.: Int. J. Methodol. 13(2), 161–173 (1979)CrossRefGoogle Scholar
  22. 22.
    Mokken, R.J., Heemskerk, E.M., Laan, S.: Close communication and 2-clubs in corporate networks: Europe 2010. Soc. Netw. Anal. Min. 6(1), 40:1–40:19 (2016)CrossRefGoogle Scholar
  23. 23.
    Paz, A., Moran, S.: Non deterministic polynomial optimization problems and their approximations. Theor. Comput. Sci. 15, 251–277 (1981)zbMATHCrossRefGoogle Scholar
  24. 24.
    Pirwani, I.A., Salavatipour, M.R.: A weakly robust PTAS for minimum clique partition in unit disk graphs. Algorithmica 62(3–4), 1050–1072 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Schäfer, A., Komusiewicz, C., Moser, H., Niedermeier, R.: Parameterized computational complexity of finding small-diameter subgraphs. Optim. Lett. 6(5), 883–891 (2012)MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Università degli Studi di BergamoBergamoItaly
  2. 2.Université de SherbrookeQuébecCanada

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