Stack Domination Density

There are infinite sequences of graphs {Gn} where |Gn| = n such that the minimal dominating sets for Gi × H fall into predictable patterns, in light of which γ (Gn × 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 {Gn} consists of cycles or path is explored in detail, and the domination density of the Grötzsch graph is calculated. For several other sequences {Gn}, 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.