Discussiones Mathematicae Graph Theory 27(3) (2007) 527-540

COUNTEREXAMPLE TO A CONJECTURE ON THE STRUCTURE OF BIPARTITE PARTITIONABLE GRAPHS

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

Abstract

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.

References

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