两类冠图的符号罗马控制数

The Signed Roman Domination Number of Two Classes Corona Graph

摘要: 设图G=(V,E)为一个简单无向图，若S⊆V，则记f(S)=∑_v∈sf(v)。若实值函数f:V→{-1,1,2}满足以下两个条件：1) 对于任意的顶点v∈V，均有f(N[v])≥1成立；2) 如果对任意顶点v∈V，若f(v)=-1，则存在一个与v相邻的顶点u∈V满足f(u)=2，则称该函数为图G的符号罗马控制函数。图G的符号罗马控制数定义为γ_sR(G)=min{f(V)|f为图G的一个符号罗马控制函数}。本文利用构造法及穷标法主要得到了k-正则图的冠图以及轮图的冠图的符号罗马控制数的精确值。

Abstract: Let G=(V,E) be a simple undirected graph and denote f(S)=∑_v∈sf(v) for S⊆V. A real-valued function f:V→{-1,1,2} is called a signed Roman domination function if f satisfies the conditions that 1) f(N[v])≥1 for any v∈V, and 2) every vertex v for which f(v)=-1 is adjacent to a vertex u for which is f(u)=2. The signed Roman domination number of G is γ_sR(G)=min{f(V)|f is a signed Roman domination function f of G} . In this paper, we determine exact values of the signed Roman domination number of two classes graph, such as a Corona of k-regular graph and wheel graph by constructive method and exhaustive method.