arXiv 1705.047352017-05math.CO

关于字典序积图的弱罗马控制数

On the weak Roman domination number of lexicographic product graphs

Magdalena Valveny, Hebert Pérez-Rosés, Juan A. Rodríguez-Velázquez

摘要：

一个图$G=(V,E)$的顶点$v$被称为关于一个函数$f: V \longrightarrow \{0,1,2\}$是不设防的，如果对于每个与$v$相邻的顶点$u$，有$f(v)=0$和$f(u)=0$。我们称函数$f$为弱罗马控制函数，如果对于每个$v$，使得$f(v)=0$，存在一个与$v$相邻的顶点$u$，使得$f(u)\in \{1,2\}$和由$f'(v)=1$，$f'(u)=f(u)-1$和$f'(z)=f(z)$定义的函数$f': V \longrightarrow \{0,1,2\}$，对于每个$z\in V \setminus\{u,v\}$，没有不设防的顶点。$f$的重量为$w(f)=\sum_{v\in V(G) }f(v)$。图$G$的弱罗马控制数，记作$γ_r(G)$，是$G$上所有弱罗马控制函数中的最小权。Henning和Hedetniemi [Discrete Math.266（2003）239-251]表明，计算$γ_r(G)$的问题是NP难的，即使限制在二分图或弦图上。这表明找到$γ_r(G)$的特殊类的图或获得良好的界限，这个不变。本文利用因子图的不变量，得到了字典序积图的弱罗马控制数的封闭公式和紧界。

英文摘要：

A vertex $v$ of a graph $G=(V,E)$ is said to be undefended with respect to a function $f: V \longrightarrow \{0,1,2\}$ if $f(v)=0$ and $f(u)=0$ for every vertex $u$ adjacent to $v$. We call the function $f$ a weak Roman dominating function if for every $v$ such that $f(v)=0$ there exists a vertex $u$ adjacent to $v$ such that $f(u)\in \{1,2\}$ and the function $f': V \longrightarrow \{0,1,2\}$ defined by $f'(v)=1$, $f'(u)=f(u)-1$ and $f'(z)=f(z)$ for every $z\in V \setminus\{u,v\}$, has no undefended vertices. The weight of $f$ is $w(f)=\sum_{v\in V(G) }f(v)$. The weak Roman domination number of a graph $G$, denoted by $γ_r(G)$, is the minimum weight among all weak Roman dominating functions on $G$. Henning and Hedetniemi [Discrete Math. 266 (2003) 239-251] showed that the problem of computing $γ_r(G)$ is NP-Hard, even when restricted to bipartite or chordal graphs. This suggests finding $γ_r(G)$ for special classes of graphs or obtaining good bounds on this invariant. In this article, we obtain closed formulae and tight bounds for the weak Roman domination number of lexicographic product graphs in terms of invariants of the factor graphs involved in the product.

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。

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英文摘要：

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