| 郝国亮,曾淑婷,庄蔚,谢智红.全局3-彩虹控制数与3-彩虹控制数之差为2和3的树的刻画[J].,2023,63(5):544-550 |
| 全局3-彩虹控制数与3-彩虹控制数之差为2和3的树的刻画 |
| Characterization of trees with difference of their global 3-rainbow domination number and 3-rainbow domination number being 2 and 3 |
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| DOI：10.7511/dllgxb202305014 |
| 中文关键词:3-彩虹控制全局3-彩虹控制补图刻画 |
| 英文关键词:3-rainbow dominationglobal 3-rainbow dominationcomplement graphcharacterization |
| 基金项目:国家自然科学基金资助项目(12061007,11861011). |
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| 摘要点击次数:109 |
| 全文下载次数:129 |
| 中文摘要: |
| 对于任意正整数k，图G的k-彩虹控制函数f定义为从图G的顶点集V到集合{1，2，…，k}的幂集的映射，使得任意满足f(u)=?的顶点u，都有∪x∈N(u)f(x)={1，2，…，k}，其中N(u)是u的开邻域．图G的k-彩虹控制函数f的权为∑x∈V(G)|f(x)|．如果f是图G及其补图的k-彩虹控制函数，则称f是图G的全局k-彩虹控制函数．图G的k-彩虹控制数γrk(G)和全局k-彩虹控制数γgrk(G)分别指图G的所有k-彩虹控制函数和所有全局k-彩虹控制函数的最小权． 2016年，Amjadi等刻画了γgr2(T)－γr2(T)=1和γgr2(T)－γr2(T)=2成立的所有树T．在此基础上，通过对图的结构分析，利用分类讨论法完全刻画了γgr3(T)－γr3(T)=2和γgr3(T)－γr3(T)=3成立的所有树T，推广了Amjadi等的结果． |
| 英文摘要: |
| For any positive integer k, k-rainbow dominating function f of graphG is a mapping from the vertex set Vto the power set of set {1,2,…,k} such that any vertex u with f(u)=? satisfies that ∪x∈N(u)f(x)={1,2,…,k}, where N(u) is the open neighborhood of u. The weight ofk-rainbow dominating function f of graph Gis ∑x∈V(G)|f(x)|. If fis ak-rainbow dominating function of graph G and its complement graph, then f is called a globalk-rainbow dominating function of graph G. The k-rainbow domination number γrk(G) and global k-rainbow domination number γgrk(G) are the minimum weights ofk-rainbow dominating function and global k-rainbow dominating function of graph G, respectively. In 2016, Amjadi et al. characterized all trees Tfor which γgr2(T)－γr2(T)=1 and γgr2(T)－γr2(T)=2. Based on above results, by classifying the structure of the graphs into different cases, all trees T for which γgr3(T)－γr3(T)=2 and γgr3(T)－γr3(T)=3 are completely characterized, which generalizes the result of Amjadi et al. |
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