IDEAS home Printed from https://ideas.repec.org/a/wsi/apjorx/v40y2023i05ns0217595923400158.html
   My bibliography  Save this article

The Extended Dominating Sets in Graphs

Author

Listed:
  • Zhipeng Gao

    (Center for Combinatorics and LPMC, Nankai University, Tianjin 300071, P. R. China)

  • Yongtang Shi

    (Center for Combinatorics and LPMC, Nankai University, Tianjin 300071, P. R. China)

  • Changqing Xi

    (Center for Combinatorics and LPMC, Nankai University, Tianjin 300071, P. R. China)

  • Jun Yue

    (School of Mathematics Science, Tiangong University, Tianjin 300387, P. R. China)

Abstract

Let G = (V (G),E(G)) be a graph and let k be an integer. A vertex subset S ⊆ V (G) is called a k-extended dominating set if every vertex u of G satisfies one of the following conditions: the distance between u and S is at most one or there are at least k different vertices s1,s2,…,sk ∈ S such that the distance between u and si (i ∈ [k]) is two. The k-extended domination number γek(G) of G is the minimum size over all k-extended dominating sets in G. When k = 2, they are called the extended dominating set and the extended domination number of G, respectively. In this paper, we mainly study the bounds of the extended domination numbers of graphs. First, we obtain the exact values of the extended domination numbers for paths and cycles. And then the Nordhaus–Gaddum bounds for the extended domination numbers are provided. Additionally, we give some bounds of the extended domination numbers for planar graphs with small diameters. Finally, we consider the k-extended domination numbers in Random graphs.

Suggested Citation

  • Zhipeng Gao & Yongtang Shi & Changqing Xi & Jun Yue, 2023. "The Extended Dominating Sets in Graphs," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 40(05), pages 1-20, October.
    • Handle: RePEc:wsi:apjorx:v:40:y:2023:i:05:n:s0217595923400158
    DOI: 10.1142/S0217595923400158
    as

    Download full text from publisher

    File URL: http://www.worldscientific.com/doi/abs/10.1142/S0217595923400158
    Download Restriction: Access to full text is restricted to subscribers
    File URL: https://libkey.io/10.1142/S0217595923400158?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:wsi:apjorx:v:40:y:2023:i:05:n:s0217595923400158. 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: Tai Tone Lim (email available below). General contact details of provider: http://www.worldscinet.com/apjor/apjor.shtml .

    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.