@article{Zhao2024, 
author = {Weisheng Zhao and Ying Li and Ruizhi Lin},
title = {The existence of a graph whose vertex set can be partitioned into a fixed number of strong domination-critical vertex-sets},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {1},
pages = {1926-1938},
keywords = {domination, critical vertex, strong critical vertex-set, vertex-critical graph},
url = {https://www.sciopen.com/article/10.3934/math.2024095},
doi = {10.3934/math.2024095},
abstract = {Let    γ  (  G  ) denote the domination number of a graph    G. A vertex    v  ∈  V  (  G  ) is called a critical vertex of    G if    γ  (  G  −  v  )  =  γ  (  G  )  −  1. A graph is called vertex-critical if its every vertex is critical. In this paper, we correspondingly introduce two such definitions: (i) A set    S  ⊆  V  (  G  ) is called a strong critical vertex-set of    G if    γ  (  G  −  S  )  =  γ  (  G  )  −      |    S      |  ; (ii) A graph    G is called strong    l-vertex-set-critical if    V  (  G  ) can be partitioned into    l strong critical vertex-sets of    G. Therefrom, we give some properties of strong    l-vertex-set-critical graphs by extending the previous results of vertex-critical graphs. As the core work, we study on the existence of this class of graphs and prove that there exists a strong    l-vertex-set-critical connected graph if and only if    l  ∉  {  2  ,  3  ,  5  }.}
}