Let $G$ be a nontrivial edge-colored connected graph. An edge-cut $R$ of $G$ is called a {\it rainbow edge-cut} if no two edges of $R$ are colored with the same color. For two distinct vertices $u$ and $v$ of $G$, if an edge-cut separates them, then the edge-cut is called a {\it $u$-$v$-edge-cut}. An edge-colored graph $G$ is called \emph{strong rainbow disconnected} if for every two distinct vertices $u$ and $v$ of $G$, there exists a both rainbow and minimum $u$-$v$-edge-cut ({\it rainbow minimum $u$-$v$-edge-cut} for short) in $G$, separating them, and this edge-coloring is called a {\it strong rainbow disconnection coloring} (srd-{\it coloring} for short) of $G$. For a connected graph $G$, the \emph{strong rainbow disconnection number} (srd-{\it number} for short) of $G$, denoted by $\textnormal{srd}(G)$, is the smallest number of colors that are needed in order to make $G$ strong rainbow disconnected. In this paper, we first characterize the graphs with $m$ edges such that $\textnormal{srd}(G)=k$ for each $k \in \{1,2,m\}$, respectively, and we also show that the srd-number of a nontrivial connected graph $G$ equals the maximum srd-number among the blocks of $G$. Secondly, we study the srd-numbers for the complete $k$-partite graphs, $k$-edge-connected $k$-regular graphs and grid graphs. Finally, we show that for a connected graph $G$, to compute $\textnormal{srd}(G)$ is NP-hard. In particular, we show that it is already NP-complete to decide if $\textnormal{srd}(G)=3$ for a connected cubic graph. Moreover, we show that for a given edge-colored (with an unbounded number of colors) connected graph $G$ it is NP-complete to decide whether $G$ is strong rainbow disconnected.

中文翻译：

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图中的强彩虹断开

令 $G$ 是一个非平凡的边着色连通图。如果没有 $R$ 的两条边是相同颜色的，则 $G$ 的边缘切割 $R$ 被称为 {\it Rainbow edge-cut}。对于$G$的两个不同的顶点$u$和$v$，如果边切割将它们分开，则边切割称为{\it $u$-$v$-edge-cut}。如果对于 $G$ 的每两个不同的顶点 $u$ 和 $v$，同时存在彩虹和最小值 $u$-$v$-，则边色图 $G$ 被称为 \emph{strong Rainbow disconnected} $G$中的边缘切割（简称{\​​it Rainbow minimum $u$-$v$-edge-cut}），将它们分开，这种边缘着色被称为{\it strong Rainbow disconnecting coloring}（srd -{\it coloring} 简称）$G$。对于连通图$G$，$G$的\emph{强彩虹断开数}（简称srd-{\it number}），记为$\textnormal{srd}(G)$，是为了使 $G$ 强彩虹断开连接所需的最少颜色数。在本文中，我们首先表征具有 $m$ 边的图，使得 $\textnormal{srd}(G)=k$ 分别对应于每个 $k \in \{1,2,m\}$，并且我们还表明非平凡连通图 $G$ 的 srd-number 等于 $G$ 块中的最大 srd-number。其次，我们研究完整的 $k$-partite 图、$k$-edge-connected $k$-regular 图和网格图的 srd-numbers。最后，我们证明对于连通图 $G$，计算 $\textnormal{srd}(G)$ 是 NP-hard。特别是，我们表明，对于连通三次图，确定 $\textnormal{srd}(G)=3$ 已经是 NP 完全的。而且，