alpha-Domination of Generalized Petersen Graph

• Main Authors: Cheng,Yi-Jie, 鄭伊婕
• Other Authors: Fu,Hung-Lin
• Format: Others
• Language: en_US
• Published: 2014
• Online Access: http://ndltd.ncl.edu.tw/handle/27779682628490463135

碩士 === 國立交通大學 === 應用數學系所 === 102 === Let G = (V,E) be a graph with n vertices, m edges and no isolated vertices. For some α with 0 < α ≤ 1 and a set S ⊆ V, we say that S is α−dominating if for all v ∈ V − S, |N(v)∩ S| ≥ α|N(v)|. The size of a smallest such S is called the α−domination number of G denoted by γα(G). For positive integers n and k, the generalized Petersen graph P(n, k) is the graph with vertex set V = {u0, u1, . . ., un−1}∪{v0, v1, . . ., vn−1} and the edge set E = {uiui+1, uivi, vivi+k | i ∈ Zn} where addition is modulo n. Clearly, P(n, k) is a 3-regular graph. In this thesis, we study γα(P(n, k)). Since for 3-regular graphs γα(G) = γ(G)(domination number of G), provided 0 < α ≤ 1/3 and γα(G) = α0(G)(vertex cover number of G) provided 2/3 < α ≤ 1, we shall focus on the case 1/3 < α ≤ 2/3. As a consequence, the exact values of γα(P(n, k)) are obtained for certain n and k.