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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 r2 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.

open journal of discrete mathematics

A Lemma on Almost Regular Graphs and an Alternative Proof for Bounds on &amp;lt;i&amp;gt;&amp;amp;#947&amp;lt;sub&amp;gt;t&amp;lt;/sub&amp;gt; (P&amp;lt;sub&amp;gt;k&amp;lt;/sub&amp;gt; □ P&amp;lt;sub&amp;gt;m&amp;lt;/sub&amp;gt;)&amp;lt;/i&amp;gt;