A Lemma on Almost Regular Graphs and an Alternative Proof for Bounds on γt (Pk □ Pm) ()
ABSTRACT
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.
Share and Cite:
P. Feit, "A Lemma on Almost Regular Graphs and an Alternative Proof for Bounds on
γt (Pk □ Pm),"
Open Journal of Discrete Mathematics, Vol. 3 No. 4, 2013, pp. 175-182. doi:
10.4236/ojdm.2013.34031.
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