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On the 2-Club Polytope of Graphs

Author

Listed:
  • Foad Mahdavi Pajouh

    (Department of Management Science and Information Systems, University of Massachusetts Boston, Boston, Massachusetts 02125)

  • Balabhaskar Balasundaram

    (School of Industrial Engineering and Management, Oklahoma State University, Stillwater, Oklahoma 74078)

  • Illya V. Hicks

    (Department of Computational and Applied Mathematics, Rice University, Houston, Texas 77005)

Abstract

A k -club is a subset of vertices of a graph that induces a subgraph of diameter at most k , where k is a positive integer. By definition, 1-clubs are cliques and the model is a distance-based relaxation of the clique definition for larger values of k . The k -club model is particularly interesting to study from a polyhedral perspective as the property is not hereditary on induced subgraphs when k is larger than one. This article introduces a new family of facet-defining inequalities for the 2-club polytope that unifies all previously known facets through a less restrictive combinatorial property, namely, independent (distance) 2-domination. The complexity of separation over this new family of inequalities is shown to be NP-hard. An exact formulation of this separation problem and a greedy separation heuristic are also proposed. The polytope described by the new inequalities (and nonnegativity) is then investigated and shown to be integral for acyclic graphs. An additional family of facets is also demonstrated for cycles of length indivisible by three. The effectiveness of these new facets as cutting planes and the difficulty of solving the separation problem in practice are then investigated via computational experiments on a test bed of benchmark instances.

Suggested Citation

  • 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.
    • Handle: RePEc:inm:oropre:v:64:y:2016:i:6:p:1466-1481
    DOI: 10.1287/opre.2016.1500
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    Cited by:

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    5. S. Raghavan & Rui Zhang, 2022. "Rapid Influence Maximization on Social Networks: The Positive Influence Dominating Set Problem," INFORMS Journal on Computing, INFORMS, vol. 34(3), pages 1345-1365, May.
    6. 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.
    7. 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.

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