The concept of rainbow disconnection number of graphs was introduced by Chartrand et al. (2018). Inspired by this concept, we put forward the concepts of rainbow vertex-disconnection and proper disconnection in graphs. In this paper, we first show that it is NP-complete to decide whether a given edge-colored graph G has a proper edge-cut separating two specified vertices, even though the graph G has \(\Delta (G)=4\) or is bipartite. Then, for a graph G with \(\Delta (G)\le 3\) we show that \(pd(G)\le 2\) and distinguish the graphs with \(pd(G)=1\) and 2, respectively. We also show that it is NP-complete to decide whether a given vertex-colored graph G is rainbow vertex-disconnected, even though the graph G has \(\Delta (G)=3\) or is bipartite.

Chartrand等人介绍了图的彩虹断开数的概念。（2018）。受此概念的启发，我们提出了彩虹顶点断开和图形中适当断开的概念。在本文中，我们第一次表明，它是一个NP完全决定给定边缘色图表是否ģ具有适当的边缘切割分离两个指定点，即使图ģ具有\（\德尔塔（G）= 4 \ ）或是二方的。然后，对于具有\（\ Delta（G）\ le 3 \）的图G，我们表明\（pd（G）\ le 2 \）并用\（pd（G）= 1 \）和2来区分图， 分别。我们还表明，确定给定的顶点彩色图是否为NP完全的即使图G具有\（\ Delta（G）= 3 \）或是二部图的，G也是彩虹顶点断开的。