, Volume 29, Issue 6, pp 1689-1711
Date: 12 Sep 2012

Stack Domination Density

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Abstract

There are infinite sequences of graphs {G n } where |G n | = n such that the minimal dominating sets for G i × H fall into predictable patterns, in light of which γ (G n × H) may be nearly linear in n; the coefficient of linearity may be regarded as the average density of the dominating set in the H-fibers of the product. The specific cases where the sequence {G n } consists of cycles or path is explored in detail, and the domination density of the Grötzsch graph is calculated. For several other sequences {G n }, the limit of this density can be seen to exist; in other cases the ratio \({\frac{\gamma (G_n \times H)}{\gamma (G_n)}}\) proves to be of greater interest, and also exists for several families of graphs.