羅馬控制數的探討

The study of Roman domination number

許智雄(Zhi-Xiong Xu)

指導教授 ： 高金美

淡江大學/理學院/數學學系碩士班/碩士(2014年)

https://doi.org/10.6846/TKU.2014.00216

摘要

在一個圖G=(V,E)上， 定義一個函數 f 將V對應到{0, 1, 2}，假如f滿足每一個對應到0 的點都有一個對應到2的鄰居，函數 f 稱為羅馬控制函數。函數f的權重為圖中所有點相應的權重總和，而所有可能的羅馬控制函數中權重最小者稱為圖 G 的羅馬控制數。一個蜘蛛圖
G(k_1,k_2,k_3,…,k_t )為含有共同端點的t個路徑〖 P〗_(k_1 ), 〖 P〗_(k_2 ), …, 〖 P〗_(k_t )所形成的圖。一個一般蜘蛛圖〖 C〗_t (k_1,k_2,k_3,…,k_t )是由一個迴圈〖 C〗_t=(1,2,3,…,t) 及t個點相異的路徑的聯集，此t個路徑分別與〖 C〗_(t )交於相異的一點且與〖 C〗_(t )交於點i的路徑為〖 P〗_(k_i )。在此論文中，我們分別獲得蜘蛛圖G(k_1,k_2,k_3,…,k_t )之控制數與羅馬控制數的計算方法，進而利用蜘蛛圖之羅馬控制數得到 γ_R (C_3 (k_(1 ), k_2, k_3 )) 以及
γ_R (C_4 (k_1,k_2,k_3,k_4 ))的計算方法，同時獲得〖 γ〗_R (C_5 (n_1,n_2,n_3,n_4,n_5 ))及γ_R (C_6 (n_1,n_2,n_3,n_4,n_5,n_6 ))=γ_R (C_3 (n_1,n_2,n_3 ))+γ_R (C_3 (n_4,n_5,n_6 ))之猜想。

關鍵字

蜘蛛圖 ； 一般蜘蛛圖 ； 羅馬控制數

並列摘要

Given a graph G = (V, E). We define a function f from V to {0, 1, 2}. The function f is called a Roman dominating function on G when satisfying the condition that every vertex v_i with f(v_i)=0 must be adjacent to at least one vertex v_j with f(v_j)=2. The weight of Roman dominating function f is the sum of the weight of each vertex of G. The minimum weight of all possible Roman dominating functions on G is the Roman domination number of G, denoted by γ_R (G). A spider graph G(k_1,k_2,k_3,…,k_t ) is the union of t paths〖 P〗_(k_1 ), 〖 P〗_(k_2 ), …, 〖 P〗_(k_t )with one common end vertex. A generalized spider graph〖 C〗_t (k_1,k_2,k_3,…,k_t ) is the union of a t-cycle〖 C〗_t=(1,2,3,…,t) and t paths〖 P〗_(k_1 ), 〖 P〗_(k_2 ), …, 〖 P〗_(k_t ) where each path intersect Ct with exact one vertex and〖 P〗_(k_i ) intersect Ct at the vertex i.
In this thesis, we obtain the formula to calculate the minimum domination number and Roman domination number of each spider graph. For the Roman domination number of a generalized spider graph, we obtain the formula of
γ_R (C_3 (k_(1 ), k_2, k_3 )) and γ_R (C_4 (k_1,k_2,k_3,k_4 )) related to the Roman domination number of a spider graph. After that we give two conjectures about calculating γ_R (C_5 (n_1,n_2,n_3,n_4,n_5 )) and γ_R (C_6 (n_1,n_2,n_3,n_4,n_5,n_6 )).

並列關鍵字

spider graph ； generalized spider graph ； Roman domination number

參考文獻

[1] Erin W. Chambers, Bill Kinnersley, Noah Prince, Douglas B. West, Extremal Problems for Roman Domination ,(2009)

[3] n. Dehgardi, s. Norouzian and s. M. Sheikholeslami, bounding the domination number of a tree in terms of its annihilation number, Transactions on Combinatorics , (2013) , pp. 9-16.

[6] C. S. ReVelle, and K. E. Rosing, Defenders imperium Romanum: A classical problem in military strategy, Amer. Math. Monthly 107, (2000), pp. 585–594.

[7] I. Stewart, Defend the Roman Empire, Scientific American 281 (1999) 136-139.

[2] E. J. Cockayne, P. A. Jr. Dreyer, S. M., Hedetniemi, and S. T. Hedetniemi, Roman domination in graphs, Discrete Mathmatics 278 (2004) 11-22.