On Proper 2-Dominating Sets in Graphs
This paper introduces the concept of proper 2-dominating sets in graphs, providing a comprehensive characterisation of graphs that possess such sets. We give the necessary and sufficient conditions for a graph to have a proper 2-dominating set. Graphs with proper dominating sets can have a symmetric...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-03-01
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| Series: | Symmetry |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-8994/16/3/296 |
| Summary: | This paper introduces the concept of proper 2-dominating sets in graphs, providing a comprehensive characterisation of graphs that possess such sets. We give the necessary and sufficient conditions for a graph to have a proper 2-dominating set. Graphs with proper dominating sets can have a symmetric structure. Moreover, we estimate the bounds of the proper 2-domination number in the graphs with respect to the 2-domination and 3-domination numbers. We show that the cardinality of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>γ</mi><mover><mn>2</mn><mo>¯</mo></mover></msub></semantics></math></inline-formula>-set is greater by one at most than the cardinality of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>γ</mi><mn>2</mn></msub></semantics></math></inline-formula>-set. |
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| ISSN: | 2073-8994 |