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作者:鄧汶
作者(英文):Deng Wen
論文名稱:圖形的可移控制數
論文名稱(英文):Transferable domination number of graphs
指導教授:郭大衛
指導教授(英文):David Kuo
口試委員:吳建銘
蔡馬良
口試委員(英文):Jiann-Ming Wu
Ma-Lian Chia
學位類別:碩士
校院名稱:國立東華大學
系所名稱:應用數學系
學號:610611002
出版年(民國):109
畢業學年度:108
語文別:英文
論文頁數:17
關鍵詞:控制集控制數可移控制數方格圖
關鍵詞(英文):dominating setdomination numbertransferable domination numbergrid
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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-step
transferable 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 →^* 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, 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 1
2 Preliminary 3
3 Transferable domination number of bipartite graphs 5
4 Transferable domination number of grids 7
5 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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