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On biconnected and fragile subgraphs of low diameter

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  • Yezerska, Oleksandra
  • Mahdavi Pajouh, Foad
  • Butenko, Sergiy

Abstract

An s-club is a subset of vertices inducing a subgraph with a diameter of at most s. It is commonly used to characterize network clusters in applications for which easy reachability between group members is of high importance. In this paper, we study two special cases of the 2-club model – a biconnected 2-club, and a fragile (not biconnected) 2-club, respectively. We investigate certain properties of both models, propose a combinatorial branch-and-bound algorithm to find a maximum biconnected 2-club, and design a polynomial-time algorithm for finding a maximum fragile 2-club in a given graph. In addition, we formulate the maximum biconnected 2-club problem as a linear 0–1 program and solve this formulation by a branch-and-cut approach where some nontrivial constraints are applied in a lazy fashion. Finally, numerical results obtained using the proposed algorithms on a test-bed of randomly generated instances and real-life graphs are also provided.

Suggested Citation

  • Yezerska, Oleksandra & Mahdavi Pajouh, Foad & Butenko, Sergiy, 2017. "On biconnected and fragile subgraphs of low diameter," European Journal of Operational Research, Elsevier, vol. 263(2), pages 390-400.
    • Handle: RePEc:eee:ejores:v:263:y:2017:i:2:p:390-400
    DOI: 10.1016/j.ejor.2017.05.020
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    References listed on IDEAS

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    1. Foad Mahdavi Pajouh & Balabhaskar Balasundaram & Illya V. Hicks, 2016. "On the 2-Club Polytope of Graphs," Operations Research, INFORMS, vol. 64(6), pages 1466-1481, December.
    2. Carvalho, Filipa D. & Almeida, M. Teresa, 2011. "Upper bounds and heuristics for the 2-club problem," European Journal of Operational Research, Elsevier, vol. 210(3), pages 489-494, May.
    3. Veremyev, Alexander & Boginski, Vladimir, 2012. "Identifying large robust network clusters via new compact formulations of maximum k-club problems," European Journal of Operational Research, Elsevier, vol. 218(2), pages 316-326.
    4. Balabhaskar Balasundaram & Sergiy Butenko & Svyatoslav Trukhanov, 2005. "Novel Approaches for Analyzing Biological Networks," Journal of Combinatorial Optimization, Springer, vol. 10(1), pages 23-39, August.
    5. Bourjolly, Jean-Marie & Laporte, Gilbert & Pesant, Gilles, 2002. "An exact algorithm for the maximum k-club problem in an undirected graph," European Journal of Operational Research, Elsevier, vol. 138(1), pages 21-28, April.
    6. Robert Mokken, 1979. "Cliques, clubs and clans," Quality & Quantity: International Journal of Methodology, Springer, vol. 13(2), pages 161-173, April.
    7. Shahram Shahinpour & Sergiy Butenko, 2013. "Algorithms for the maximum k-club problem in graphs," Journal of Combinatorial Optimization, Springer, vol. 26(3), pages 520-554, October.
    8. R. Luce & Albert Perry, 1949. "A method of matrix analysis of group structure," Psychometrika, Springer;The Psychometric Society, vol. 14(2), pages 95-116, June.
    9. Pattillo, Jeffrey & Youssef, Nataly & Butenko, Sergiy, 2013. "On clique relaxation models in network analysis," European Journal of Operational Research, Elsevier, vol. 226(1), pages 9-18.
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    Citations

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    Cited by:

    1. Foad Mahdavi Pajouh, 2020. "Minimum cost edge blocker clique problem," Annals of Operations Research, Springer, vol. 294(1), pages 345-376, November.
    2. Komusiewicz, Christian & Nichterlein, André & Niedermeier, Rolf & Picker, Marten, 2019. "Exact algorithms for finding well-connected 2-clubs in sparse real-world graphs: Theory and experiments," European Journal of Operational Research, Elsevier, vol. 275(3), pages 846-864.
    3. Veremyev, Alexander & Boginski, Vladimir & Pasiliao, Eduardo L. & Prokopyev, Oleg A., 2022. "On integer programming models for the maximum 2-club problem and its robust generalizations in sparse graphs," European Journal of Operational Research, Elsevier, vol. 297(1), pages 86-101.
    4. Zhong, Haonan & Mahdavi Pajouh, Foad & Prokopyev, Oleg A., 2021. "Finding influential groups in networked systems: The most degree-central clique problem," Omega, Elsevier, vol. 101(C).
    5. Yajun Lu & Hosseinali Salemi & Balabhaskar Balasundaram & Austin Buchanan, 2022. "On Fault-Tolerant Low-Diameter Clusters in Graphs," INFORMS Journal on Computing, INFORMS, vol. 34(6), pages 3181-3199, November.
    6. Niels Grüttemeier & Philipp Heinrich Keßler & Christian Komusiewicz & Frank Sommer, 2024. "Efficient branch-and-bound algorithms for finding triangle-constrained 2-clubs," Journal of Combinatorial Optimization, Springer, vol. 48(3), pages 1-27, October.

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