论文标题
关于digraphs的总彩虹统治
On the total Rainbow domination of digraphs
论文作者
论文摘要
对于一个正整数$ k $，digraph $ d $上的$ k $ rainbow主导功能（$ k $ rdf）是从顶点套装$ f $ from the the the the $ f $ from $ f $ to $ \ {1,2，\ ldots，\ ldots，k \} $ for vertex $ v $ v $ f fortex $ v $ f（v） $ \ bigCup_ {u \ in n^ - （v）} f（u）= \ {1,2，\ ldots，k \} $，其中$ n^ - （v）$是$ v $的neighbors集合。 $ k $ rdf $ f $的重量定义为$ \ sum_ {v \ in v（d）} | f（v）| $。 $ k $ rdf $ f $ in $ d $没有孤立的顶点称为$ k $ rainbow主导功能，如果套装$ \ {v \ in v（d）：f（v）：f（v）\ ne \ ne \ emptyset \} $的子数字$ d $的子数据被称为$ d $。总$ k $ - 李子弓的统治数是$ d $上$ k $ rainbow统治功能的最小重量。在本文中，我们为总$ k $ rainbow的统治编号建立了一些界限，并给出了总$ k $ - rainbow统治数量的一些挖掘机。
For a positive integer $k$, a $k$-rainbow dominating function ($k$RDF) on a digraph $D$ is a function $f$ from the vertex set $V(D)$ to the set of all subsets of $\{1,2,\ldots,k\}$ such that for any vertex $v$ with $f(v)=\emptyset$, $\bigcup_{u\in N^-(v)}f(u)=\{1,2,\ldots,k\}$, where $N^-(v)$ is the set of in-neighbors of $v$. The weight of a $k$RDF $f$ is defined as $\sum_{v\in V(D)}|f(v)|$. A $k$RDF $f$ on $D$ with no isolated vertex is called a total $k$-rainbow dominating function if the subdigraph of $D$ induced by the set $\{v\in V(D):f(v)\ne\emptyset\}$ has no isolated vertex. The total $k$-rainbow domination number is the minimum weight of a total $k$-rainbow dominating function on $D$. In this paper, we establish some bounds for the total $k$-rainbow domination number and we give the total $k$-rainbow domination number of some digraphs.
