Discussiones Mathematicae Graph Theory 27(3) (2007) 527-540
doi: 10.7151/dmgt.1377

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Richard G. Gibson  and  Christina M. Mynhardt

Department of Mathematics and Statistics
University of Victoria
P.O. Box 3045, Victoria, BC Canada V8W 3P4
e-mail: richardg@sfu.ca,  mynhardt@math.uvic.ca


A graph G is called a prism fixer if γ(G×K2) = γ(G), where γ(G) denotes the domination number of G. A symmetric γ-set of G is a minimum dominating set D which admits a partition D = D1∪D2 such that V(G)−N[Di] = Dj, i,j = 1,2, i ≠ j. It is known that G is a prism fixer if and only if G has a symmetric γ-set.

Hartnell and Rall [On dominating the Cartesian product of a graph and K2, Discuss. Math. Graph Theory 24 (2004), 389-402] conjectured that if G is a connected, bipartite graph such that V(G) can be partitioned into symmetric γ-sets, then G ≅ C4 or G can be obtained from K2t,2t by removing the edges of t vertex-disjoint 4-cycles. We construct a counterexample to this conjecture and prove an alternative result on the structure of such bipartite graphs.

Keywords: domination, prism fixer, symmetric dominating set, bipartite graph.

2000 Mathematics Subject Classification: 05C69.


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Received 21 August 2006
Revised 21 February 2007
Accepted 7 March 2007