## Characterizing Cartesian Fixers and Multipliers

 Stephen Benecke and Christina M. Mynhardt Department of Mathematics and Statistics University of Victoria, P.O. Box 3060 STN CSC Victoria, B.C., Canada V8W 3R4

## Abstract

Let G☐ H denote the Cartesian product of the graphs G and H. In 2004, Hartnell and Rall [On dominating the Cartesian product of a graph and K2, Discuss. Math. Graph Theory 24(3) (2004), 389-402] characterized prism fixers, i.e., graphs G for which γ(G☐ K2) = γ(G), and noted that γ(G☐ Kn) ≥ min{ |V(G) |, γ(G)+n−2}. We call a graph G a consistent fixer if γ(G☐ Kn) = γ(G)+n−2 for each n such that 2 ≤ n < |V(G) |− γ(G)+2, and characterize this class of graphs.

Also in 2004, Burger, Mynhardt and Weakley [On the domination number of prisms of graphs, Dicuss. Math. Graph Theory 24(2) (2004), 303-318] characterized prism doublers, i.e., graphs G for which γ(G☐K2) = 2 γ(G). In general γ(G☐ Kn) ≤ n γ(G) for any n ≥ 2. We call a graph attaining equality in this bound a Cartesian n-multiplier and also characterize this class of graphs.

Keywords: Cartesian product, prism fixer, Cartesian fixer, prism doubler, Cartesian multiplier, domination number

2010 Mathematics Subject Classification: 05C69, 05C99.

## References

 [1] A.P. Burger, C.M. Mynhardt and W.D. Weakley, On the domination number of prisms of graphs, Dicuss. Math. Graph Theory 24 (2004) 303--318, doi: 10.7151/dmgt.1233. [2] G. Chartrand and F. Harary, Planar permutation graphs, Ann. Inst. H. Poincaré Sect. B (N.S.) 3 (1967) 433--438. [3] B.L. Hartnell and D.F. Rall, Lower bounds for dominating Cartesian products, J. Combin. Math. Combin. Comput. 31 (1999) 219--226. [4] B.L. Hartnell and D.F. Rall, On dominating the Cartesian product of a graph and K2, Discuss. Math. Graph Theory 24 (2004) 389--402, doi: 10.7151/dmgt.1238. [5] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998). [6] C.M. Mynhardt and Z. Xu, Domination in prisms of graphs: Universal fixers, Utilitas Math. 78 (2009) 185--201.

Received 26 February 2009
Revised 15 March 2011
Accepted 4 April 2011