## PAIRED DOMINATION IN PRISMS OF GRAPHS

Christina M. Mynhardt and Mark Schurch

Department of Mathematics and Statistics
University of Victoria
P.O. Box 3060 STN CSC
e-mail: mynhardt@math.uvic.ca, mschurch@math.uvic.ca

## Abstract

The paired domination number γpr(G) of a graph G is the smallest cardinality of a dominating set S of G such that áSñ has a perfect matching. The generalized prisms πG of G are the graphs obtained by joining the vertices of two disjoint copies of G by |V(G)| independent edges. We provide characterizations of the following three classes of graphs: γpr(πG) = 2γpr(G) for all πG; γpr(K2□G) = 2γpr(G); γpr(K2□G) = γpr(G).

Keywords: domination, paired domination, prism of a graph, Cartesian product.

2010 Mathematics Subject Classification: 05C69.

## References

 [1] B. Bresar, M.A. Henning and D.F. Rall, Paired-domination of Cartesian products of graphs, Util. Math. 73 (2007) 255-265. [2] A.P. Burger and C.M. Mynhardt, Regular graphs are not universal fixers, Discrete Math. 310 (2010) 364-368, doi: 10.1016/j.disc.2008.09.016. [3] A.P. Burger, C.M. Mynhardt and W.D. Weakley, On the domination number of prisms of graphs, Discuss. Math. Graph Theory 24 (2004) 303-318, doi: 10.7151/dmgt.1233. [4] E.J. Cockayne, R.G. Gibson and C.M. Mynhardt, Claw-free graphs are not universal fixers, Discrete Math. 309 (2009) 128-133, doi: 10.1016/j.disc.2007.12.053. [5] M. Edwards, R.G. Gibson, M.A. Henning and C.M. Mynhardt, On paired-domination edge critical graphs, Australasian J. Combin. 40 (2008) 279-292. [6] R.G. Gibson, Bipartite graphs are not universal fixers, Discrete Math. 308 (2008) 5937-5943, doi: 10.1016/j.disc.2007.11.006. [7] B.L. Hartnell and D.F. Rall, On Vizing's conjecture, Congr. Numer. 82 (1991) 87-96. [8] 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. [9] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998). [10] C.M. Mynhardt and Z. Xu, Domination in prisms of graphs: Universal fixers, Utilitas Math. 78 (2009) 185-201. [11] M. Schurch, Domination Parameters for Prisms of Graphs (Master's thesis, University of Victoria, 2005).