Complexity results for the proper disconnection of graphs
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For an edge-colored graph G, a set F of edges of G is called a proper edge-cut if F is an edge-cut of G and any pair of adjacent edges in F are assigned by different colors. An edge-colored graph is called proper disconnected if for each pair of distinct vertices of G there exists a proper edge-cut separating them. For a connected graph G, the proper disconnection number of G, denoted by pd(G), is defined as the minimum number of colors that are needed to make G proper disconnected. In this paper, we first show that it is NP-complete to decide whether a given k-edge-colored graph G with Δ(G)=4 is proper disconnected. Then, for a graph G with Δ(G)≤ 3 we show that pd(G)≤ 2 and determine the graphs with pd(G)=1 and 2, respectively. Finally, we show that for a general graph G, deciding whether pd(G)=1 is NP-complete, even if G is bipartite.
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