@article{Fu2023, 
author = {Miao Fu and Yuqin Zhang},
title = {Results on monochromatic vertex disconnection of graphs},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {6},
pages = {13219-13240},
keywords = {monochromatic vertex cut, monochromatic vertex disconnection number, connectivity, block, Erdős-Gallai-type problems},
url = {https://www.sciopen.com/article/10.3934/math.2023668},
doi = {10.3934/math.2023668},
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 have a monochromatic vertex cut separating them. The monochromatic vertex disconnection number of    G, denoted by    m  v  d  (  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:    m  v  d  (  G  ), connectivity and block decomposition. We determine the value of    m  v  d  (  G  ) for some well-known graphs, and then characterize    G when    n  −  5  ≤  m  v  d  (  G  )  ≤  n and all blocks of    G are minimally 2-connected triangle-free graphs. We obtain the maximum size of a graph    G with    m  v  d  (  G  )  =  k for any    k. Finally, we study the Erdős-Gallai-type results for    m  v  d  (  G  ), and completely solve them.}
}