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Finding large \(k\)-clubs in undirected graphs

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Abstract

Finding cohesive subgroups is an important issue in studying social networks. Many models exist for defining cohesive subgraphs in social networks, such as clique, \(k\)-clique, and \(k\)-clan. The concept of \(k\)-club is one of them. A \(k\)-club of a graph is a maximal subset of the vertex set which induces a subgraph of diameter \(k\). It is a relaxation of a clique, which induces a subgraph of diameter \(1\). We conducted algorithmic studies on finding a \(k\)-club of size as large as possible. In this paper, we show that one can find a \(k\)-club of maximum size in \(O^{*}(1.62^n)\) time where \(n\) is the number of vertices of the input graph. We implemented a combinatorial branch-and-bound algorithm that finds a \(k\)-club of maximum size and a new heuristic algorithm called IDROP given in this paper. To speed up the programs, we introduce a dynamic data structure called \(k\)-DN which, under deletion of vertices from a graph, maintains for a given vertex \(v\) the set of vertices at distances at most \(k\). From the experimental results that we obtained, we concluded that a \(k\)-club of maximum size can be easily found in sparse graphs and dense graphs. Our heuristic algorithm finds, within reasonable time, \(k\)-clubs of maximum size in most of experimental instances. The gap between the size of a \(k\)-club of maximum size and a \(k\)-club found by IDROP is a constant for the number of vertices that we are able to test.

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Notes

  1. The source code can be download at the link

    https://sites.google.com/site/kdynamicneighborhoodproject/.

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Acknowledgments

We would like to thank anonymous reviewers for their valuable comments and suggestions which were very useful in improving the presentation of this paper.

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Correspondence to Ling-Ju Hung.

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This research is partially supported by the National Science Council of Taiwan under grants NSC 98–2221–E–241–018–MY3, NSC 99–2221–E–241–015–MY3, and NSC 101–2221–E–241–019–MY3.

Part of this paper has been presented in Proceedings of the 28th Workshop on Combinatorial Mathematics and Computation Theory.

L.-J. Hung is supported by the National Science Council of Taiwan under grant NSC 101–2811–E–241–002.

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Chang, MS., Hung, LJ., Lin, CR. et al. Finding large \(k\)-clubs in undirected graphs. Computing 95, 739–758 (2013). https://doi.org/10.1007/s00607-012-0263-3

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