On domination number of Cartesian product of directed cycles

Abstract

Let γ ( G ) denote the domination number of a digraph G and let C m □ C n denote the Cartesian product of C m and C n , the directed cycles of length m , n ⩾ 2 . In this paper, we determine the exact values: γ ( C 2 □ C n ) = n ; γ ( C 3 □ C n ) = n if n ≡ 0 ( mod 3 ) , otherwise, γ ( C 3 □ C n ) = n + 1 ; γ ( C 4 □ C n ) = 3 n 2 if n ≡ 0 ( mod 8 ) , otherwise, γ ( C 4 □ C n ) = n + ⌈ n + 1 2 ⌉ .