Graphs and Combinatorics

, Volume 29, Issue 6, pp 1689–1711 | Cite as

Stack Domination Density

  • Timothy Brauch
  • Paul Horn
  • Adam Jobson
  • D. Jacob Wildstrom
Original Paper
  • 91 Downloads

Abstract

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.

Keywords

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

Mathematics Subject Classification (2000)

05C69 05C76 

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Copyright information

© Springer 2012

Authors and Affiliations

  • Timothy Brauch
    • 1
  • Paul Horn
    • 2
  • Adam Jobson
    • 3
  • D. Jacob Wildstrom
    • 3
  1. 1.Department of MathematicsManchester UniversityNorth ManchesterUSA
  2. 2.Department of MathematicsHarvard UniversityCambridgeUSA
  3. 3.Department of MathematicsUniversity of LouisvilleLouisvilleUSA

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