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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Acknowledgements
The authors are very grateful to the reviewers and editor for their helpful suggestions and comments. This paper is an extended version of Chen et al. (2020), which was published in the proceedings of FAW 2020, Lecture Notes in Computer Science No.12340.
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This study was supported by NSFC No. 11871034.
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Chen, Y., Li, P., Li, X. et al. Complexity results for two kinds of colored disconnections of graphs. J Comb Optim 42, 40–55 (2021). https://doi.org/10.1007/s10878-021-00742-0
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DOI: https://doi.org/10.1007/s10878-021-00742-0