A vertex-colored graph is rainbow vertex-connected if any two vertices are connected by a path whose internal vertices have distinct colors, which was introduced by Krivelevich and Yuster. The rainbow vertex-connection of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex-connected. In this paper, we study the computational complexity of vertex-rainbow connection of graphs and prove that computing rvc(G) is NP-Hard. Moreover, we show that it is already NP-Complete to decide whether rvc(G) = 2. We also prove that the following problem is NP-Complete: given a vertex-colored graph G, check whether the given coloring makes G rainbow vertexconnected.
A vertex-colored graph G is rainbow vertex-connected if any pair of distinct vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection number of G, denoted by rvc(G), is the minimum number of colors that are needed to make G rainbow vertex-connected. In this paper we give a Nordhaus-Gaddumtype result of the rainbow vertex-connection number. We prove that when G and G are both connected, then 2 ≤ rvc(G) + rvc(G) ≤ n − 1.Examples are given to show that both the upper bound and the lower bound are best possible for all n ≥ 5.
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