Abstract
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.
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