Let G be a vertex-colored graph. We call a vertex cut S of G a monochromatic vertex cut if the vertices of S are colored with the same color. The 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, we propose an algorithm to compute mvd(G) and give an mvd-coloring of the graph G. We run this algorithm with an example written in Java. The main part of the code is shown in Appendix B and the complete code is given on Github: https://github.com/fumiaoT/mvd-coloring.git. Secondly, inspired by the previous localization principle, we obtain a upper bound of mvd(G) for some special classes of graphs. In addition, when mvd(G) is large and all blocks of G are minimally 2-connected trianglefree graphs, we characterize G. On these bases, we show that any graph whose blocks are all minimally 2-connected graphs of small order, can be computed mvd(G) and given an mvd-coloring in polynomial time.
<abstract><p>Let $ G $ be a vertex-colored graph. A vertex cut $ S $ of $ G $ is called a <italic>monochromatic vertex cut</italic> if the vertices of $ S $ are colored with the same color. A graph $ G $ is <italic>monochromatically vertex-disconnected</italic> if any two nonadjacent vertices of $ G $ have a monochromatic vertex cut separating them. The <italic>monochromatic vertex disconnection number</italic> 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 $. Finally, we study the Erdős-Gallai-type results for $ mvd(G) $, and completely solve them.</p></abstract>
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