arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1105.3290math.CO
收藏

图的Roman控制与Mycieleki结构

Roman domination and Mycieleki's structure in graphs

Adel P. Kazemi

发表信息：
Ars Combinatorics, Volume CVI, July 2012 pp. 277-287
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

对于图$G=(V,E)$，如果对于任意一个具有$f(v)=0$的顶点$v$，在其邻域内至少有一个具有$f(w)=2$的顶点$w$，则称一个函数$f:V\rightarrow \{0,1,2\}$为罗马控制函数（RDF）. $G$的RDF $f$的权重是值$f(V)=\sum_{v\in V}f(v)$。$G$的RDF的最小权是它的罗马控制数，记为$γ_ R(G)$。本文首先证明了$γ_{R}(G)+1\leq γ_{R}(μ(G))\leq γ_{R}(G)+2$，其中$μ(G)$是$G$的Mycielekian图，然后刻画了在这些界中达到相等的图.然后，对任意正整数$m$，计算了特殊罗马图$G$的$m$-Mycieleskian $μ_{m}(G)$在$γ_R(G)$上的罗马控制数.最后，我们提出了几个图表来说明讨论的图形。

英文摘要：

For a graph $G=(V,E)$, a function $f:V\rightarrow \{0,1,2\}$ is called Roman dominating function (RDF) if for any vertex $v$ with $f(v)=0$, there is at least one vertex $w$ in its neighborhood with $f(w)=2$. The weight of an RDF $f$ of $G$ is the value $f(V)=\sum_{v\in V}f(v)$. The minimum weight of an RDF of $G$ is its Roman domination number and denoted by $γ_ R(G)$. In this paper, we first show that $γ_{R}(G)+1\leq γ_{R}(μ(G))\leq γ_{R}(G)+2$, where $μ(G)$ is the Mycielekian graph of $G$, and then characterize the graphs achieving equality in these bounds. Then for any positive integer $m$, we compute the Roman domination number of the $m$-Mycieleskian $μ_{m}(G)$ of a special Roman graph $G$ in terms on $γ_R(G)$. Finally we present several graphs to illustrate the discussed graphs.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑