## Abstract

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 D_{1} is single-step transferable to D_{2} if there exist u∈D_{1} and v∈D_{2}, such that uv∈E(G) and D_{1}−{u}=D_{2}−{v}. We write D_{1}⟶∗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}⟶∗D_{2} for any D_{1},D_{2}∈D(G) with |D_{1}|=|D_{2}|=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.

Original language | English |
---|---|

Pages (from-to) | 135-146 |

Number of pages | 12 |

Journal | Discrete Applied Mathematics |

Volume | 313 |

DOIs | |

State | Published - 31 May 2022 |

## Keywords

- Dominating set
- Domination number
- Grid
- Transferable domination number