  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 triangle-free 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.
