Monochromatic vertex-disconnection of graphs

TL;DR

This paper introduces the concept of monochromatic vertex-disconnection number in graphs, explores its relationship with graph parameters, characterizes specific graph classes, and provides algorithms for computing and coloring based on this parameter.

Contribution

It defines the monochromatic vertex-disconnection number, studies its properties, characterizes graphs with high mvd(G), and presents algorithms for its computation and coloring.

Findings

Determined mvd(G) for well-known graphs.

Characterized graphs with high mvd(G) and specific block structures.

Developed an algorithm to compute mvd(G) and produce an mvd-coloring.

Abstract

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 Erd\H{o}s-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.