Let $G$ be a nontrivial connected and vertex colored graph. A vertex cut $S$ of $G$ is called a rainbow vertex cut if no two vertices of it are colored the same. The graph $G$ is called \emph{rainbow vertex disconnected} if for any two nonadjacent vertices $x$ and $y$, there exists an $x-y$ rainbow vertex cut. We introduce and study the \emph{rainbow vertex disconnection number} $rvd(G)$ of a connected graph $G$, which is defined as the minimum number of colors that are needed to make $G$ rainbow vertex disconnected. In this paper, we first characterize the connected graphs $G$ with $rvd(G)=1$ and $n$, respectively. We also characterize the minimally $2$-connected graphs $G$ for which $rvd(G)=2$ and $n-2$, respectively. Secondly, we determine the rainbow vertex disconnection numbers for complete multipartite graphs and grid graphs. Finally, we get the maximum size of a connected graph $G$ of order $n$ with $rvd(G)=k$ for given integers $k$ and $n$ with $1\leq k\leq n$.

中文翻译：

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图中的彩虹顶点断开

让 $G$ 是一个非平凡的连接和顶点彩色图。如果没有两个顶点颜色相同，则 $G$ 的顶点切割 $S$ 称为彩虹顶点切割。如果对于任何两个不相邻的顶点 $x$ 和 $y$，存在 $xy$ 彩虹顶点切割，则图 $G$ 称为 \emph{rainbow vertex disconnected}。我们介绍并研究了连通图$G$的\emph{rainbow vertex disconnection number} $rvd(G)$，它被定义为使$G$彩虹顶点断开所需的最小颜色数。在本文中，我们首先分别刻画 $rvd(G)=1$ 和 $n$ 的连通图 $G$。我们还分别刻画了 $rvd(G)=2$ 和 $n-2$ 的最小 $2$-connected 图 $G$。其次，我们确定完整多部图和网格图的彩虹顶点断开数。最后，