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Let γ(G) denote the domination number of a graph G and let Cn□G denote the cartesian product of Cn, the cycle of length n⩾3, and G. In this paper, we are mainly concerned with the question: which connected nontrivial graphs satisfy γ(Cn□G)=γ(Cn)γ(G)? We prove that this equality can only hold if n≡1 (mod3). In addition, we characterize graphs which satisfy this equality when n=4 and provide infinite...