Abstract
Covering a graph with cohesive subgraphs is a classical problem in theoretical computer science. In this paper, we prove new complexity results on the
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
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
on some graph classes. We prove that
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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Recall that a matching is a set of edges that share no endpoint.
References
Alba, R.D.: A graph-theoretic definition of a sociometric clique. J. Math. Sociol. 3, 113–126 (1973)
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)
Balasundaram, B., Butenko, S., Trukhanov, S.: Novel approaches for analyzing biological networks. J. Comb. Optim. 10(1), 23–39 (2005)
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)
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)
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)
Chang, M., Hung, L., Lin, C., Su, P.: Finding large k-clubs in undirected graphs. Computing 95(9), 739–758 (2013)
Dondi, R., Mauri, G., Sikora, F., Zoppis, I.: Covering a graph with clubs. J. Graph Algorithms Appl. 23(2), 271–292 (2019)
Dondi, R., Mauri, G., Zoppis, I.: On the tractability of finding disjoint clubs in a network. Theor. Comput. Sci. (2019, to appear)
Dumitrescu, A., Pach, J.: Minimum clique partition in unit disk graphs. Graphs Comb. 27(3), 399–411 (2011)
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)
Garey, M.R., Johnson, D.S., Stockmeyer, L.J.: Some simplified NP-complete graph problems. Theor. Comput. Sci. 1(3), 237–267 (1976)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)
Golovach, P.A., Heggernes, P., Kratsch, D., Rafiey, A.: Finding clubs in graph classes. Discrete Appl. Math. 174, 57–65 (2014)
Hartung, S., Komusiewicz, C., Nichterlein, A.: Parameterized algorithmics and computational experiments for finding 2-clubs. J. Graph Algorithms Appl. 19(1), 155–190 (2015)
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)
Kloks, T.: Treewidth, Computations and Approximations. LNCS, vol. 842. Springer, Heidelberg (1994). https://doi.org/10.1007/BFb0045375
Komusiewicz, C.: Multivariate algorithmics for finding cohesive subnetworks. Algorithms 9(1), 21 (2016)
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)
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)
Mokken, R.: Cliques, clubs and clans. Qual. Quant.: Int. J. Methodol. 13(2), 161–173 (1979)
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)
Paz, A., Moran, S.: Non deterministic polynomial optimization problems and their approximations. Theor. Comput. Sci. 15, 251–277 (1981)
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)
Schäfer, A., Komusiewicz, C., Moser, H., Niedermeier, R.: Parameterized computational complexity of finding small-diameter subgraphs. Optim. Lett. 6(5), 883–891 (2012)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Dondi, R., Lafond, M. (2019). On the Tractability of Covering a Graph with 2-Clubs. In: Gąsieniec, L., Jansson, J., Levcopoulos, C. (eds) Fundamentals of Computation Theory. FCT 2019. Lecture Notes in Computer Science(), vol 11651. Springer, Cham. https://doi.org/10.1007/978-3-030-25027-0_17
Download citation
DOI: https://doi.org/10.1007/978-3-030-25027-0_17
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-25026-3
Online ISBN: 978-3-030-25027-0
eBook Packages: Computer ScienceComputer Science (R0)