Transferable domination number of graphs

Fei Huang Chang, Ma Lian Chia, David Kuo, Wen Deng, Sheng Chyang Liaw, Zhishi Pan

Research output: Contribution to journalArticlepeer-review


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 Pm×Pn 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 languageEnglish
Pages (from-to)135-146
Number of pages12
JournalDiscrete Applied Mathematics
StatePublished - 31 May 2022


  • Dominating set
  • Domination number
  • Grid
  • Transferable domination number


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