Gravier et al. established bounds on the size of a minimal totally dominant subset for graphs Pk□Pm. This paper offers an alternative calculation, based on the following lemma: Let so k≥3 and r≥2. Let H be an r-reg...Gravier et al. established bounds on the size of a minimal totally dominant subset for graphs Pk□Pm. 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=Pk□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 Pk□ Cn when k and n satisfy specific modular conditions. Bounds for rt(Pk□Pm) , 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.展开更多
文摘Gravier et al. established bounds on the size of a minimal totally dominant subset for graphs Pk□Pm. 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=Pk□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 Pk□ Cn when k and n satisfy specific modular conditions. Bounds for rt(Pk□Pm) , 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.
