IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v441y2023ics0096300322007858.html
   My bibliography  Save this article

Plane graphs of diameter two are 2-optimal

Author

Listed:
  • Kong, Jiangxu
  • Wang, Yiqiao
  • Hu, Jiacheng
  • Wang, Yang
  • Wang, Weifan

Abstract

Let G be a connected graph with n vertices. Suppose a fire breaks out at some vertex of G. At each time interval, firefighters can protect up to k vertices, and then the fire spreads to all unprotected neighbors. Let dnk(v) denote the minimum number of vertices that the fire may damage when a fire breaks out at vertex v. The k-expected damage of G, denoted by εk(G), is the expectation of the proportion of vertices that can be damaged from the fire, if the starting vertex of the fire is chosen uniformly at random, i.e., εk(G)=∑v∈V(G)dnk(v)/n2. A class of graphs G is called k-optimal if εk(G) tends to 0 as n tends to infinity for any G∈G. In this paper, we prove that planar graphs of diameter two are 2-optimal, which is the best possible.

Suggested Citation

  • Kong, Jiangxu & Wang, Yiqiao & Hu, Jiacheng & Wang, Yang & Wang, Weifan, 2023. "Plane graphs of diameter two are 2-optimal," Applied Mathematics and Computation, Elsevier, vol. 441(C).
    • Handle: RePEc:eee:apmaco:v:441:y:2023:i:c:s0096300322007858
    DOI: 10.1016/j.amc.2022.127717
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300322007858
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2022.127717?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.

    References listed on IDEAS

    as
    1. Weifan Wang & Stephen Finbow & Ping Wang, 2014. "A lower bound of the surviving rate of a planar graph with girth at least seven," Journal of Combinatorial Optimization, Springer, vol. 27(4), pages 621-642, May.
    2. Shu-Yu Cui & Yiqiao Wang & Danjun Huang & Hongwei Du & Weifan Wang, 2022. "Structure and pancyclicity of maximal planar graphs with diameter two," Journal of Combinatorial Optimization, Springer, vol. 43(1), pages 1-27, January.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. O. Zambon, Mauricio J. & J. de Rezende, Pedro & C. de Souza, Cid, 2019. "Solving the geometric firefighter routing problem via integer programming," European Journal of Operational Research, Elsevier, vol. 274(3), pages 1090-1101.
    2. Tingting Wu & Jiangxu Kong & Weifan Wang, 2016. "The 2-surviving rate of planar graphs without 5-cycles," Journal of Combinatorial Optimization, Springer, vol. 31(4), pages 1479-1492, May.

    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:eee:apmaco:v:441:y:2023:i:c:s0096300322007858. 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.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with 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: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    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.