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.
Conflicts of Interest
The authors declare no conflicts of interest.
References
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[1]
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S. Gravier, “Total Domination of Grid Graphs,” Discrete Applied Mathematics, Vol. 121, No. 1-3, 2002, pp. 119-128. .
http://dx.doi.org/10.1016/S0166-218X(01)00297-9
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[2]
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S. Gravier, M. Molland and C. Payan, “Variations on Tilings in Manhattan Metric,” Geometriae Dedicata, Vol. 76, No. 3, 1999, pp. 265-273.
http://dx.doi.org/10.1023/A:1005106901394
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