Transferable domination number of graphs

Let G be a connected graph, and let D(G) be the set of all dominating (multi)sets for G. For D₁ and D₂ in D(G), we say that D₁ is single-step transferable to D₂ if there exist u∈D₁ and v∈D₂, such that uv∈E(G) and D₁−{u}=D₂−{v}. We write D₁⟶∗D₂ if D₁ can be transferred to D₂ through a sequence of single-step transfers. We say that G is k-transferable if D₁⟶∗D₂ for any D₁,D₂∈D(G) with |D₁|=|D₂|=k. The transferable domination number of G is the smallest integer k to guarantee that G is l-transferable for all l≥k. We study the transferable domination number of graphs in this paper. We give upper bounds for the transferable domination number of general graphs and bipartite graphs, and give a lower bound for the transferable domination number of grids. We also determine the transferable domination number of P_m×P_n for the cases that m=2,3, or mn≡0(mod6). Besides these, we give an example to show that the gap between the transferable domination number of a graph G and the smallest number k so that G is k-transferable can be arbitrarily large.