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


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.


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

Mathematics Subject Classification (2000)

05C69 05C76 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bayati, M., Gamarnik, D., Tetali, P.: Combinatorial approach to the interpolation method and scaling limits in sparse random graphs. In: STOC’10—Proceedings of the 2010 ACM International Symposium on Theory of Computing, pp. 105–114. ACM, New York (2010)Google Scholar
  2. 2.
    Chang T.Y., Clark W.E., Hare E.O.: Domination numbers of complete grid graphs. I. Ars Combin. 38, 97–111 (1994)MathSciNetMATHGoogle Scholar
  3. 3.
    Fisher D.C., McKenna P.A., Boyer E.D.: Hamiltonicity, diameter, domination, packing, and biclique partitions of Mycielski’s graphs. Discrete Appl. Math. 84(1–3), 93–105 (1998). doi:10.1016/S0166-218X(97)00126-1 MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Golay M.J.E.: Notes on digital coding. Proceedings of the IRE 37(6), 657 (1949). doi:10.1109/JRPROC.1949.233620 Google Scholar
  5. 5.
    Jacobson M.S., Kinch L.F.: On the domination number of products of graphs. I. Ars Combin. 18, 33–44 (1984)MathSciNetMATHGoogle Scholar
  6. 6.
    Jacobson M.S., Kinch L.F.: On the domination of the products of graphs. II. Trees. J. Graph Theory 10(1), 97–106 (1986). doi:10.1002/jgt.3190100112 MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    King E.L.C., Pelsmajer M.J.: Dominating sets in plane triangulations. Discrete Math. 310(17–18), 2221–2230 (2010). doi:10.1016/j.disc.2010.03.022 MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Liu H., Pelsmajer M.J.: Dominating sets in triangulations on surfaces. Ars Math. Contemp. 4(1), 177–204 (2011)MathSciNetMATHGoogle Scholar
  9. 9.
    Molloy M., Reed B.: Graph Colouring and the Probabilistic Method. Springer, Berlin (2002)CrossRefMATHGoogle Scholar
  10. 10.
    Rubalcaba, R., Slater, P.J.: Efficient cartesian product layer domination. In: 22nd Cumberland Conference on Combinatorics, Graph Theory and Computing (2009)Google Scholar
  11. 11.
    Vizing V.G.: Some unsolved problems in graph theory. Uspehi Mat. Nauk 23(6 (144), 117–134 (1968)MathSciNetMATHGoogle Scholar

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

Personalised recommendations