Dominion on Grids

Jianning Su, Julian Allagan (), Shanzhen Gao, Olumide Malomo, Weizheng Gao and Ephrem Eyob
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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

Mathematics, 2024, vol. 12, issue 21, 1-13

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.

Keywords: domination; dominion; grids (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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