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Monochromatic vertex-disconnection of graphs

Let G be a vertex-colored graph. A vertex cut S of G is called a monochromatic vertex cut if the vertices of S are colored with the same color. A graph G is monochromatically vertex-disconnected if any two nonadjacent vertices of G has a monochromatic vertex cut separating them. The monochromatic vertex-disconnection number of G, denoted by mvd(G), is the maximum number of colors that are used to make G monochromatically vertex-disconnected. In this paper, the connection between the graph parameters are studied: mvd(G), connectivity and block decomposition. We determine the value of mvd(G) for some well known graphs, and then characterize G when n-5\leq mvd(G)\leq n and all blocks of G are minimally 2-connected triangle-free graphs. We obtain the maximum size of a graph G with mvd(G)=k for any k. Furthermore, we study the Erdős-Gallai-type results for mvd(G), and completely solve them. Finally, we propose an algorithm to compute mvd(G) and give an mvd-coloring of G.