On the Domination Number of Cartesian Product of Two Directed Cycles

Denote by gamma(G) the domination number of a digraph G and C-m square C-n the Cartesian product of C-m and C-n, the directed cycles of length m, n >= 2. In 2010, Liu et al. determined the exact values of gamma(C-m square C-n) for m = 2, 3, 4, 5, 6. In 2013, Mollard determined the exact values of gamma(C-m square C-n) for m = 3k + 2. In this paper, we give lower and upper bounds of gamma(C-m square C-n) with m = 3k + 1 for different cases. In particular, inverted right perpendicular(2k + 1)n/2inverted left perpendicular <= gamma(C3k+1 square C-n) <= left perpendicular(2k + 1)n/2right perpendicular + k Based on the established result, the exact values of gamma(C-m square C-n) are determined for m = 7 and 10 by the combination of the dynamic algorithm, and an upper bound for gamma(C-13 square C-n) is provided.