Counterexample to a Conjecture on the Structure of Bipartite Partitionable Graphs

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 6= 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.