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The existence of a graph whose vertex set can be partitioned into a fixed number of strong domination-critical vertex-sets
- Source :
- AIMS Mathematics, Vol 9, Iss 1, Pp 1926-1938 (2024)
- Publication Year :
- 2024
- Publisher :
- AIMS Press, 2024.
-
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\} $.
Details
- Language :
- English
- ISSN :
- 24736988
- Volume :
- 9
- Issue :
- 1
- Database :
- Directory of Open Access Journals
- Journal :
- AIMS Mathematics
- Publication Type :
- Academic Journal
- Accession number :
- edsdoj.9219f4c802e74c0eb46f0144381a7c8c
- Document Type :
- article
- Full Text :
- https://doi.org/10.3934/math.2024095?viewType=HTML