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- W3045862158 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 $Delta(G)=4$ is proper disconnected. Then, for a graph $G$ with $Delta(G)leq 3$ 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 $Delta(G)=3$ or is bipartite." @default.
- W3045862158 created "2020-08-03" @default.
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- W3045862158 date "2019-12-21" @default.
- W3045862158 modified "2023-10-16" @default.
- W3045862158 title "Complexity results for two kinds of colored disconnections of graphs" @default.
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- W3045862158 doi "https://doi.org/10.48550/arxiv.1912.10349" @default.
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