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 令G是一個連通圖且D(G)是對於一個G收集所有控制(多重)集的集合。對於D1,D2∈D(G)，如果存在u∈D1和v∈D2，使得uv∈E(G)且D1−{u}=D2−{v}，則我們稱為D1 是單步可轉移為D2，並記為 D1 → D2。而如果D1是可以經由多步(一個序列)轉移為D2，則記為 D1 →^* D2。如果 D1 →^* D2，對於D1,D2∈D(G)且|D1|=|D2|=k.，則稱G是k可轉移的。圖形G的可移控制數是指最小整數k，對於所有的 l ≥ k，可以保證G為l可轉移的，我們用γt∗ (G) 來表示圖形 G 的可移控制數。本文研究了圖形的可移控制數，我們給出了二部圖的可移控制數的上界、方格圖的可移控制數的下界，並且我們還確定 P2 × Pn 和 P3 × Pn 的可移控制數。除此之外，我們舉一個例子來說明，圖形G的可移控制數與最小整數k之間的差距可以任意大，而使得G為k可轉移的。
 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-steptransferable to D2, denoted as D1 → D2, if there exist u ∈ D1 and v ∈ D2, such that uv∈E(G)and D1−{u}=D2−{v}. We write D1 →^* D2if 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, denoted by γt∗(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 thesis. We give an upper bound for the transferable domination number of bipartite graphs and give a lower bound for the transferable domination number of grids. We also determine the transferable domination number of P2 × Pn and P3 × Pn. 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.
 1 Introduction 12 Preliminary 33 Transferable domination number of bipartite graphs 54 Transferable domination number of grids 75 The gap between the smallest number to guarantee mutual transferability and the transferable domination number 15
 [1] K. T. Chu, W. H. Lin, C. Chen, Mutual transferability for (F, B, R)-domination on strongly chordal graphs and cactus graphs, Discrete Appl. Math. 259 (2019) 41–52.[2] J. F. Fink, M. S. Jacobson, L. F. Kinch, J. Roberts, The bondage number of a graph, Discrete Math. 86 (1990) 47–57.[3] S. Fujita, A tight bound on the number of mobile servers to guarantee transferability among dominating configurations, Discrete Appl. Math. 158 (2010) 913–920.[4] D. Gonc ̧alves, A. Pinlou, M. Rao, S. Thomass ́e, The domination number of grids, SIAM J. Discrete Math. 25 (2011) 1443–1453.[5] T. W. Haynes, S. T. Hedetniemi, P. J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, New York, 1998.[6] T. W. Haynes, S. T. Hedetniemi, P. J. Slater (Eds.), Domination in Graphs: Advanced Topics, Marcel Dekker, New York, 1998.
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