论文标题
外部独立$ 2 $ rainbow主导函数的进一步结果
Further results on outer independent $2$-rainbow dominating functions of graphs
论文作者
论文摘要
令$ g =（v（g），e（g））$为图。 a函数$ f：v（g）\ rightarrow \ mathbb {p}（\ {1,2 \}）$是$ 2 $ -RAINBOW主导函数，如果对于每个顶点$ v $ at $ f（v）= \ emberySet $，v $ v $ v $ $ g $的外部独立$ 2 $ -RAINBOW主导功能（OI $ 2 $ RD功能）是$ 2 $ - rainbow统治函数$ f $，其中所有$ v \ in v（g）$ in $ f（v）= \ emptySet $的集合是独立的。外部独立$ 2 $ -RAINBOW统治号码（oi $ 2 $ rd号）$γ_{oir2}（g）$是$ g $的oi $ 2 $ rd函数的最小重量。 在本文中，我们首先证明$ n/2 $是OI $ 2 $ rd的下限，订单$ n $的无连接爪图的数量，并表征了平等所持的所有此类图，解决了早期论文中给出的空旷问题。此外，还对某些图产品进行了此参数的研究。特别是，我们给出了一个封闭的（分别是）oi $ 2 $ rd的植根（分子电晕）产品图的封闭公式，并在此参数上证明了笛卡尔产品的上限和两个图的直接乘积。
Let $G=(V(G),E(G))$ be a graph. A function $f:V(G)\rightarrow \mathbb{P}(\{1,2\})$ is a $2$-rainbow dominating function if for every vertex $v$ with $f(v)=\emptyset$, $f\big{(}N(v)\big{)}=\{1,2\}$. An outer-independent $2$-rainbow dominating function (OI$2$RD function) of $G$ is a $2$-rainbow dominating function $f$ for which the set of all $v\in V(G)$ with $f(v)=\emptyset$ is independent. The outer independent $2$-rainbow domination number (OI$2$RD number) $γ_{oir2}(G)$ is the minimum weight of an OI$2$RD function of $G$. In this paper, we first prove that $n/2$ is a lower bound on the OI$2$RD number of a connected claw-free graph of order $n$ and characterize all such graphs for which the equality holds, solving an open problem given in an earlier paper. In addition, a study of this parameter for some graph products is carried out. In particular, we give a closed (resp. an exact) formula for the OI$2$RD number of rooted (resp. corona) product graphs and prove upper bounds on this parameter for the Cartesian product and direct product of two graphs.
