TITLE:
A Lemma on Almost Regular Graphs and an Alternative Proof for Bounds on γt (Pk □ Pm)
AUTHORS:
Paul Feit
KEYWORDS:
Domination; Total Domination; Matrix; Linear Algebra
JOURNAL NAME:
Open Journal of Discrete Mathematics,
Vol.3 No.4,
October
24,
2013
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.