The existence of a graph whose vertex set can be partitioned into a fixed number of strong domination-critical vertex-sets

Abstract

Let $ \gamma(G) $ denote the domination number of a graph $ G $. A vertex $ v\in V(G) $ is called a critical vertex of $ G $ if $ \gamma(G-v) = \gamma(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\subseteq V(G) $ is called a strong critical vertex-set of $ G $ if $ \gamma(G-S) = \gamma(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\notin\{2, 3, 5\} $.

Keywords

• domination
• critical vertex
• strong critical vertex-set
• vertex-critical graph