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 D-1 and D-2 in D(G), we say that D1 is single-step transferable to D-2 if there exist u is an element of D-1 and v is an element of D-2, such that uv is an element of E(G) and D-1 - {u} = D-2 - {v}. We write D-1 star -> D-2 if D-1 can be transferred to D-2 through a sequence of single-step transfers. We say that G is * k-transferable if D-1 star -> D-2 for any D-1, D-2 is an element of D(G) with vertical bar D-1 vertical bar = vertical bar D-2 vertical bar = 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 Pm x Pn for the cases that m = 2, 3, or mn = 0 (mod 6). 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. (c) 2022 Elsevier B.V. All rights reserved.