IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v12y2024i21p3408-d1511256.html
   My bibliography  Save this article

Dominion on Grids

Author

Listed:
  • Jianning Su

    (Department of Mathematics, Computer Science, and Engineering, Perimeter College, Georgia State University, Clarkston, GA 30021, USA)

  • Julian Allagan

    (Department of Mathematics, Computer Science and Engineering Technology, Elizabeth City State University, Elizabeth City, NC 27909, USA)

  • Shanzhen Gao

    (Department of Computer Information Systems, Virginia State University, Petersburg, VA 23806, USA)

  • Olumide Malomo

    (Department of Computer Information Systems, Virginia State University, Petersburg, VA 23806, USA)

  • Weizheng Gao

    (Department of Mathematics, Computer Science and Engineering Technology, Elizabeth City State University, Elizabeth City, NC 27909, USA)

  • Ephrem Eyob

    (Department of Computer Information Systems, Virginia State University, Petersburg, VA 23806, USA)

Abstract

The domination (number) of a graph G = ( V , E ) , denoted by γ ( G ) , is the size of the minimum dominating sets of V ( G ) , also known as γ -sets. As such, the dominion of G , denoted by ζ ( G ) , counts all its γ -sets. We proved a conjecture from one of the authors on the dominion of cycles C 3 k − 1 and C 3 k − 2 , k ≥ 2 . Further, we found the formulae and recurrence relations for the dominions of several grids, G m , n , with 2 ≤ m ≤ 4 and other results when m ≤ 9 and n ≤ 20 . In general, domination and dominion play important roles in assessing certain vulnerabilities of any given network system.

Suggested Citation

  • Jianning Su & Julian Allagan & Shanzhen Gao & Olumide Malomo & Weizheng Gao & Ephrem Eyob, 2024. "Dominion on Grids," Mathematics, MDPI, vol. 12(21), pages 1-13, October.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:21:p:3408-:d:1511256
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/12/21/3408/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/12/21/3408/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Duncan J. Watts & Steven H. Strogatz, 1998. "Collective dynamics of ‘small-world’ networks," Nature, Nature, vol. 393(6684), pages 440-442, June.
    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. Samrachana Adhikari & Beau Dabbs, 2018. "Social Network Analysis in R: A Software Review," Journal of Educational and Behavioral Statistics, , vol. 43(2), pages 225-253, April.
    2. Wang, Xiaojie & Slamu, Wushour & Guo, Wenqiang & Wang, Sixiu & Ren, Yan, 2022. "A novel semi local measure of identifying influential nodes in complex networks," Chaos, Solitons & Fractals, Elsevier, vol. 158(C).
    3. Lin, Dan & Wu, Jiajing & Xuan, Qi & Tse, Chi K., 2022. "Ethereum transaction tracking: Inferring evolution of transaction networks via link prediction," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 600(C).
    4. Ferreira, D.S.R. & Ribeiro, J. & Oliveira, P.S.L. & Pimenta, A.R. & Freitas, R.P. & Dutra, R.S. & Papa, A.R.R. & Mendes, J.F.F., 2022. "Spatiotemporal analysis of earthquake occurrence in synthetic and worldwide data," Chaos, Solitons & Fractals, Elsevier, vol. 165(P2).
    5. Qinghu Liao & Wenwen Dong & Boxin Zhao, 2023. "A New Strategy to Solve “the Tragedy of the Commons” in Sustainable Grassland Ecological Compensation: Experience from Inner Mongolia, China," Sustainability, MDPI, vol. 15(12), pages 1-24, June.
    6. Jianhong Chen & Hongcai Ma & Shan Yang, 2023. "SEIOR Rumor Propagation Model Considering Hesitating Mechanism and Different Rumor-Refuting Ways in Complex Networks," Mathematics, MDPI, vol. 11(2), pages 1-22, January.
    7. Pawanesh Pawanesh & Charu Sharma & Niteesh Sahni, 2025. "Analyzing Communicability and Connectivity in the Indian Stock Market During Crises," Papers 2502.08242, arXiv.org.
    8. Daniel Reisinger & Fabian Tschofenig & Raven Adam & Marie Lisa Kogler & Manfred Füllsack & Fabian Veider & Georg Jäger, 2024. "Patterns of stability in complex contagions," Journal of Computational Social Science, Springer, vol. 7(2), pages 1895-1911, October.
    9. Gregory Gutin & Tomohiro Hirano & Sung-Ha Hwang & Philip R. Neary & Alexis Akira Toda, 2021. "The effect of social distancing on the reach of an epidemic in social networks," Journal of Economic Interaction and Coordination, Springer;Society for Economic Science with Heterogeneous Interacting Agents, vol. 16(3), pages 629-647, July.
    10. Jie, Ke-Wei & Liu, San-Yang & Sun, Xiao-Jun & Xu, Yun-Cheng, 2023. "A dynamic ripple-spreading algorithm for solving mean–variance of shortest path model in uncertain random networks," Chaos, Solitons & Fractals, Elsevier, vol. 167(C).
    11. Yu Gong & Xiaojiang Xu & Changping Zhao & Tobias Schoenherr, 2024. "Multi-Tier Supply Chain Learning Networks: A Simulation Study Based on the Experience-Weighted Attraction (EWA) Model," Sustainability, MDPI, vol. 16(10), pages 1-25, May.
    12. Divakaruni, Anantha & Zimmerman, Peter, 2023. "The Lightning Network: Turning Bitcoin into money," Finance Research Letters, Elsevier, vol. 52(C).
    13. Leto Peel & Tiago P. Peixoto & Manlio De Domenico, 2022. "Statistical inference links data and theory in network science," Nature Communications, Nature, vol. 13(1), pages 1-15, December.
    14. Chen, Feng & Wu, Bin & Lou, Wen-qian & Zhu, Bo-wen, 2024. "Impact of dual-credit policy on diffusion of technology R & D among automakers: Based on an evolutionary game model with technology-spillover in complex network," Energy, Elsevier, vol. 303(C).
    15. Xiaodi Ni & Lijun Yang, 2024. "Mapping Salience and Trajectory: On How to Situate Literary Translators in Publishing Legends of the Condor Heroes With Visualization," SAGE Open, , vol. 14(2), pages 21582440241, May.
    16. Abderrahim Zannou & Abdelhak Boulaalam & El Habib Nfaoui, 2020. "SIoT: A New Strategy to Improve the Network Lifetime with an Efficient Search Process," Future Internet, MDPI, vol. 13(1), pages 1-23, December.
    17. Jingsha He & Yue Li & Nafei Zhu, 2023. "A Game Theory-Based Model for the Dissemination of Privacy Information in Online Social Networks," Future Internet, MDPI, vol. 15(3), pages 1-17, February.
    18. Qian, Qian & Feng, Hairong & Gu, Jing, 2021. "The influence of risk attitude on credit risk contagion—Perspective of information dissemination," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 582(C).
    19. Fangyuan Tian & Hongxia Li & Shuicheng Tian & Chenning Tian & Jiang Shao, 2022. "Is There a Difference in Brain Functional Connectivity between Chinese Coal Mine Workers Who Have Engaged in Unsafe Behavior and Those Who Have Not?," IJERPH, MDPI, vol. 19(1), pages 1-21, January.
    20. Yelai Feng & Huaixi Wang & Chao Chang & Hongyi Lu, 2022. "Intrinsic Correlation with Betweenness Centrality and Distribution of Shortest Paths," Mathematics, MDPI, vol. 10(14), pages 1-18, July.

    More about this item

    Keywords

    domination; dominion; grids;
    All these keywords.

    Statistics

    Access and download statistics

    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:gam:jmathe:v:12:y:2024:i:21:p:3408-:d:1511256. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.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.