Dominion on Grids

The domination (number) of a graph G=(V,E), denoted by gamma(G), is the size of the minimum dominating sets of V(G), also known as gamma-sets. As such, the dominion of G, denoted by zeta(G), counts all its gamma-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.