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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
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    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.
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    Keywords

    domination; dominion; grids;
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