Abstract
Clubs are generalizations of cliques. For a positive integer s, an s-club in a graph G is a set of vertices that induces a subgraph of G of diameter at most s. The importance and fame of cliques are evident, whereas clubs provide more realistic models for practical applications. Computing an s-club of maximum cardinality is an NP-hard problem for every fixed s ≥ 1, and this problem has attracted significant attention recently. We present new positive results for the problem on large and important graph classes. In particular we show that for input G and s, a maximum s-club in G can be computed in polynomial time when G is a chordal bipartite or a strongly chordal or a distance hereditary graph. On a superclass of these graphs, weakly chordal graphs, we obtain a polynomial-time algorithm when s is an odd integer, which is best possible as the problem is NP-hard on this class for even values of s. We complement these results by proving the NP-hardness of the problem for every fixed s on 4-chordal graphs, a superclass of weakly chordal graphs. Finally, if G is an AT-free graph, we prove that the problem can be solved in polynomial time when s ≥ 2, which gives an interesting contrast to the fact that the problem is NP-hard for s = 1 on this graph class.
This work is supported by the European Research Council and the Research Council of Norway.
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References
Abello, J., Pardalos, P., Resende, M.: On maximum clique problems in very large graphs. In: External memory algorithms and visualization. DIMACS Series on Discrete Mathematics and Theoretical Computer Science, vol. 50, pp. 119–130. AMS (1999)
Agnarsson, G., Greenlaw, R., Halldórsson, M.M.: On powers of chordal graphs and their colorings. In: Proceedings of SICCGTC 2000, vol. 144, pp. 41–65 (2000)
Alba, R.: A graph-theoretic definition of a sociometric clique. Journal of Mathematical Sociology 3, 113–126 (1973)
Asahiro, Y., Miyano, E., Samizo, K.: Approximating maximum diameter-bounded subgraphs. In: López-Ortiz, A. (ed.) LATIN 2010. LNCS, vol. 6034, pp. 615–626. Springer, Heidelberg (2010)
Balakrishnan, R., Paulraja, P.: Correction to: “Graphs whose squares are chordal”. Indian J. Pure Appl. Math. 12(8), 1062 (1981)
Balakrishnan, R., Paulraja, P.: Graphs whose squares are chordal. Indian J. Pure Appl. Math. 12(2), 193–194 (1981)
Balasundaram, B., Butenko, S., Trukhanov, S.: Novel approaches for analyzing biological networks. Journal of Combinatorial Optimization 10, 23–39 (2005)
Bandelt, H.J., Henkmann, A., Nicolai, F.: Powers of distance-hereditary graphs. Discrete Mathematics 145(1-3), 37–60 (1995)
Brandstädt, A., Le, V., Spinrad, J.: Graph Classes: A Survey. SIAM Monographs on Discrete Mathematics and Applications (1999)
Brandstädt, A., Dragan, F.F., Xiang, Y., Yan, C.: Generalized powers of graphs and their algorithmic use. In: Arge, L., Freivalds, R. (eds.) SWAT 2006. LNCS, vol. 4059, pp. 423–434. Springer, Heidelberg (2006)
Chandran, L.S., Mathew, R.: Bipartite powers of k-chordal graphs. arXiv:1108.0277 [math.CO] (2012)
Chang, J.M., Ho, C.W., Ko, M.T.: Powers of asteroidal triple-free graphs with applications. Ars Comb. 67 (2003)
Chang, M.S., Hung, L.J., Lin, C.R., Su, P.C.: Finding large k-clubs in undirected graphs. In: Proceedings of the 28th Workshop on Combinatorial Mathematics and Computation Theory, pp. 1–10 (2011)
Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem. Ann. of Math (2) 164(1), 51–229 (2006)
Corneil, D., Perl, Y.: Clustering and domination in perfect graphs. Discrete Applied Mathematics 9, 27–39 (1984)
Dawande, M., Swaminathan, J., Keskinocak, P., Tayur, S.: On bipartite and multipartite clique problems. Journal of Algorithms 41, 388–403 (2001)
Diestel, R.: Graph theory, 4th edn. Graduate Texts in Mathematics, vol. 173. Springer, Heidelberg (2010)
Golumbic, M.C.: Algorithmic graph theory and perfect graphs, 2nd edn. Annals of Discrete Mathematics, vol. 57. Elsevier (2004)
Golumbic, M.C., Rotics, U.: On the clique-width of some perfect graph classes. Int. J. Found. Comput. Sci. 11(3), 423–443 (2000)
Hartung, S., Komusiewicz, C., Nichterlein, A.: Parameterized algorithmics and computational experiments for finding 2-clubs. In: Thilikos, D.M., Woeginger, G.J. (eds.) IPEC 2012. LNCS, vol. 7535, pp. 231–241. Springer, Heidelberg (2012)
Hayward, R., Hoàng, C.T., Maffray, F.: Optimizing weakly triangulated graphs. Graphs and Combinatorics 5(1), 339–349 (1989)
Hayward, R., Spinrad, J., Sritharan, R.: Weakly chordal graph algorithms via handles. In: Proceedings of SODA 2000, pp. 42–49 (2000)
Kloks, T., Kratsch, D.: Computing a perfect edge without vertex elimination ordering of a chordal bipartite graph. Information Processing Letters 55 (1995)
Lubiw, A.: Γ-free matrices. Master thesis, Department of Combinatorics and Optimization, University of Waterloo (1982)
Luce, R.: Connectivity and generalized cliques in sociometric group structure. Psychometrika 15, 169–190 (1950)
Mokken, R.: Cliques, clubs and clans. Quality and Quantity 13, 161–173 (1979)
Pajouh, F.M., Balasundaram, B.: On inclusionwise maximal and maximum cardinality k-clubs in graphs. Discrete Optimization 9(2), 84–97 (2012)
Poljak, S.: A note on stable sets and colorings of graphs. Comment. Math. Univ. Carolinae 15, 307–309 (1974)
Schäfer, A.: Exact algorithms for s-club finding and related problems (2009), diplomarbeit, Institut für Informatik, Friedrich-Schiller-Universität Jena
Schäfer, A., Komusiewicz, C., Moser, H., Niedermeier, R.: Parameterized computational complexity of finding small-diameter subgraphs. Optimization Letters 6(5), 883–891 (2012)
Seidman, S.B., Foster, B.L.: A graph theoretic generalization of the clique concept. Journal of Mathematical Sociology 6, 139–154 (1978)
Tarjan, R.E., Yannakakis, M.: Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM J. Comput. 13(3), 566–579 (1984)
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Golovach, P.A., Heggernes, P., Kratsch, D., Rafiey, A. (2013). Cliques and Clubs. In: Spirakis, P.G., Serna, M. (eds) Algorithms and Complexity. CIAC 2013. Lecture Notes in Computer Science, vol 7878. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38233-8_23
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DOI: https://doi.org/10.1007/978-3-642-38233-8_23
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