IDEAS home Printed from https://ideas.repec.org/a/spr/annopr/v349y2025i3d10.1007_s10479-025-06571-2.html
   My bibliography  Save this article

Minimum-cardinality global defensive alliances in general graphs

Author

Listed:
  • André Rossi

    (Université Paris Dauphine - PSL)

  • Alok Singh

    (University of Hyderabad)

Abstract

A subset S of vertices of an undirected graph G is a defensive alliance if at least half of the vertices in the closed neighborhood of each vertex of S are in S. A defensive alliance is a global defensive alliance if it is also a dominating set of G. This paper addresses the problem of finding minimum-cardinality global defensive alliances for general graphs. Two integer linear programming formulations are proposed to address this problem, the second one being an improved version of the first one in which the constraints are attempted for tightening with a cubing-time algorithm. Two new lower bounds on the cardinality of a defensive global alliance are proposed: the first one is based on a linear time algorithm and is shown to be tighter than three of the four lower bounds from the literature, and the second one is derived from the linear programming relaxation of the aforementioned integer linear programming formulations of the problem. An upper bound on the global defensive alliance number is obtained using a greedy peeling algorithm that is shown to be at least as good as an upper bound of the literature, however it is also shown that the proposed algorithm may be unable to find an optimal solution for some graphs. Finally, numerical experiments are carried out on the 78 DIMACS instances and on 75 Erdős-Rényi graphs with up to 10,000 vertices in order to show the effectiveness of the proposed approaches.

Suggested Citation

  • André Rossi & Alok Singh, 2025. "Minimum-cardinality global defensive alliances in general graphs," Annals of Operations Research, Springer, vol. 349(3), pages 1891-1931, June.
    • Handle: RePEc:spr:annopr:v:349:y:2025:i:3:d:10.1007_s10479-025-06571-2
    DOI: 10.1007/s10479-025-06571-2
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10479-025-06571-2
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10479-025-06571-2?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:annopr:v:349:y:2025:i:3:d:10.1007_s10479-025-06571-2. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.