Original Paper

Graphs and Combinatorics

, Volume 29, Issue 6, pp 1689-1711

First online:

Stack Domination Density

  • Timothy BrauchAffiliated withDepartment of Mathematics, Manchester University
  • , Paul HornAffiliated withDepartment of Mathematics, Harvard University
  • , Adam JobsonAffiliated withDepartment of Mathematics, University of Louisville
  • , D. Jacob WildstromAffiliated withDepartment of Mathematics, University of Louisville Email author 

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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.


Domination number Cartesian product Grötzsch graph Asymptotic density Additive graphs Random graphs

Mathematics Subject Classification (2000)

05C69 05C76