Results on 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 $G$ have 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 $n-5\le mvd(G)\le n$ 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.