Gravier et al. established bounds on the size of a minimal totally dominant subset for graphs P_k□P_m. This paper offers an alternative calculation, based on the following lemma: Let so k≥3 and r≥2. Let H be an r-regular finite graph, and put G=P_k□H. 1) If a perfect totally dominant subset exists for G, then it is minimal; 2) If r＞2 and a perfect totally dominant subset exists for G, then every minimal totally dominant subset of G must be perfect. Perfect dominant subsets exist for P_k□ C_n when k and n satisfy specific modular conditions. Bounds for r_t(P_k□P_m) , for all k,m follow easily from this lemma. Note: The analogue to this result, in which we replace “totally dominant” by simply “dominant”, is also true.