Dominion on Grids
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 C3k−1 and C3k−2, k≥2. Further, we found the formulae and recurrence relations for the dominions of several grids, Gm,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.
Authors
- Shanzhen Gao (ORCID: https://orcid.org/0000-0002-3856-2530)
- Ephrem Eyob (ORCID: https://orcid.org/0000-0003-3590-0028)
- Jianning Su (ORCID: https://orcid.org/0000-0002-7443-8940)
- Julian D. Allagan (ORCID: https://orcid.org/0000-0003-0275-4239)
- Olumide Malomo (ORCID: https://orcid.org/0000-0003-2592-6953)
- Weizheng Gao (ORCID: https://orcid.org/0009-0003-5078-6283)
Institutions
- Virginia State University (US)
- Georgia State University (US)
- Elizabeth City State University (US)
Publication Details
- Journal
- Mathematics
- Published
- 2024-10-31
- DOI
- https://doi.org/10.3390/math12213408
- Citations
- 3
- Primary Topic
- Advanced Graph Theory Research
- Type
- article
- Field-Weighted Citation Impact
- 1.08