Algorithm for monochromatic vertex-disconnection of graphs

TL;DR

This paper introduces an algorithm to compute the monochromatic vertex-disconnection number of a graph, providing practical implementation, bounds for specific graph classes, and polynomial-time computation methods for certain block-structured graphs.

Contribution

It proposes a new algorithm for calculating mvd(G), offers bounds for special graph classes, and characterizes graphs with large mvd(G) when blocks are minimally 2-connected triangle-free.

Findings

Algorithm successfully computes mvd(G) for various graphs.

Provides upper bounds for mvd(G) in specific classes.

Polynomial-time computation for graphs with certain block structures.

Abstract

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.