Conflict-free connections: algorithm and complexity
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A path in an(a) edge(vertex)-colored graph is called a conflict-free path if there exists a color used on only one of the edges(vertices) of the path. An(A) edge(vertex)-colored graph is called conflict-free (vertex-)connected if between each pair of distinct vertices of the graph there is a conflict-free path. We call the graph strongly conflict-free connected if between each pair of distinct vertices of the graph there exists a conflict-free shortest path. The strong conflict-free connection number of a connected graph G, denoted by scfc(G), is defined as the smallest number of colors that are required to make G strongly conflict-free connected. In this paper, we study the problem: Given a connected graph G and a coloring c: E(or V)→{1,2,...,k} (k≥ 1) of G, determine whether or not G is conflict-free (vertex-)connected under the coloring c. We provide a polynomial-time algorithm for this problem. We then show that it is NP-complete to decide whether there is a 2-edge-coloring of G=(V,E) such that, for a given subset P of V× V, all pairs (u,v)∈ P are strongly conflict-free connected. At last, we show that it is NP-hard to decide whether scfc(G)≤ k (k≥ 3) for a given graph G.
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