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arXiv · 1912.10349

Complexity results for two kinds of colored disconnections of graphs

You Chen·Ping Li·Xueliang Li·Yindi Weng
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Abstract

The concept of rainbow disconnection number of graphs was introduced by Chartrand et al. in 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 $G$ with maximum degree $Δ(G)=4$ is proper disconnected. Then, for a graph $G$ with $Δ(G)\leq 3$ we show that $pd(G)\leq 2$ and determine the graphs with $pd(G)=1$ and $2$, respectively. Furthermore, we show that for a general graph $G$, deciding whether $pd(G)=1$ is $NP$-complete, even if $G$ is bipartite. 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 $Δ(G)=3$ or is bipartite.

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You Chen, Ping Li, Xueliang Li, Yindi Weng. 2020-07-29. Complexity results for two kinds of colored disconnections of graphs. https://arxiv.org/abs/1912.10349

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