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Complexity results for two kinds of colored disconnections of graphs

Author

Listed:
  • You Chen

    (Nankai University)

  • Ping Li

    (Nankai University)

  • Xueliang Li

    (Nankai University)

  • Yindi Weng

    (Nankai University)

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 $$\Delta (G)=4$$ Δ ( G ) = 4 or is bipartite. Then, for a graph G with $$\Delta (G)\le 3$$ Δ ( G ) ≤ 3 we show that $$pd(G)\le 2$$ p d ( G ) ≤ 2 and distinguish the graphs with $$pd(G)=1$$ p d ( 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 $$\Delta (G)=3$$ Δ ( G ) = 3 or is bipartite.

Suggested Citation

  • You Chen & Ping Li & Xueliang Li & Yindi Weng, 2021. "Complexity results for two kinds of colored disconnections of graphs," Journal of Combinatorial Optimization, Springer, vol. 42(1), pages 40-55, July.
    • Handle: RePEc:spr:jcomop:v:42:y:2021:i:1:d:10.1007_s10878-021-00742-0
    DOI: 10.1007/s10878-021-00742-0
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