Complexity results for two kinds of colored disconnections of graphs

TL;DR

This paper investigates the computational complexity of various disconnection concepts in edge-colored and vertex-colored graphs, establishing NP-completeness results and characterizing graphs with small proper disconnection numbers.

Contribution

It introduces the concepts of rainbow vertex-disconnection and proper disconnection, and proves NP-completeness for deciding these properties in certain classes of graphs.

Findings

Deciding proper disconnection is NP-complete for graphs with maximum degree 4.

Graphs with maximum degree 3 have proper disconnection number at most 2.

Deciding rainbow vertex-disconnection is NP-complete even for bipartite graphs with degree 3.

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.