Equitable and Paired Equitable Domination in Inflated Graphs and Their Complements
Domination plays an indispensable role in graph theory. Various types of domination explore various types of applications. Equal-status people work together and interlace with each other easily. In this paper, the paired equitable domination of a graph, its inflated graph, and its complement of an i...
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| Format: | Article |
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MDPI AG
2023-12-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/12/12/1134 |
| _version_ | 1797382020465164288 |
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| author | Narayanan Kumaran Annamalai Meenakshi Robert Cep Jayavelu Udaya Prakash Ondrej Mizera |
| author_facet | Narayanan Kumaran Annamalai Meenakshi Robert Cep Jayavelu Udaya Prakash Ondrej Mizera |
| author_sort | Narayanan Kumaran |
| collection | DOAJ |
| description | Domination plays an indispensable role in graph theory. Various types of domination explore various types of applications. Equal-status people work together and interlace with each other easily. In this paper, the paired equitable domination of a graph, its inflated graph, and its complement of an inflated graph were studied. The relationship between the domination number of the graph, the equitable domination number, and the paired equitable domination number of complements of the inflated graph were established. Furthermore, we proved the Nordhaus–Gaddum-type inequality, that is, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mrow><mi>γ</mi></mrow><mrow><mi>p</mi><mi>r</mi></mrow><mrow><mi>e</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>+</mo><msubsup><mrow><mi>γ</mi></mrow><mrow><mi>p</mi><mi>r</mi></mrow><mrow><mi>e</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>≤</mo><mn>6</mn></mrow></semantics></math></inline-formula> if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi></mrow></semantics></math></inline-formula> is a graph with <i>m</i> nodes where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>≡</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>2</mn><mo stretchy="false">(</mo><mi>m</mi><mi>o</mi><mi>d</mi><mo> </mo><mn>8</mn><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> = (<i>m</i>/2) for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></semantics></math></inline-formula>. The challenges and limitations of this parameter of paired equitable and equitable domination depends on the degree of the vertex of the graph. Practical applications are discussed in various fields and illustrated using the studied parameter. |
| first_indexed | 2024-03-08T20:59:27Z |
| format | Article |
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| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-08T20:59:27Z |
| publishDate | 2023-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-33e3635e774f4c66afb3d02e0e9d7cfa2023-12-22T13:53:20ZengMDPI AGAxioms2075-16802023-12-011212113410.3390/axioms12121134Equitable and Paired Equitable Domination in Inflated Graphs and Their ComplementsNarayanan Kumaran0Annamalai Meenakshi1Robert Cep2Jayavelu Udaya Prakash3Ondrej Mizera4Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai 600062, IndiaDepartment of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai 600062, IndiaDepartment of Machining, Assembly and Engineering Metrology, Faculty of Mechanical Engineering, VSB-Technical University of Ostrava, 70800 Ostrava, Czech RepublicDepartment of Mechanical Engineering, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai 600062, IndiaDepartment of Machining, Assembly and Engineering Metrology, Faculty of Mechanical Engineering, VSB-Technical University of Ostrava, 70800 Ostrava, Czech RepublicDomination plays an indispensable role in graph theory. Various types of domination explore various types of applications. Equal-status people work together and interlace with each other easily. In this paper, the paired equitable domination of a graph, its inflated graph, and its complement of an inflated graph were studied. The relationship between the domination number of the graph, the equitable domination number, and the paired equitable domination number of complements of the inflated graph were established. Furthermore, we proved the Nordhaus–Gaddum-type inequality, that is, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mrow><mi>γ</mi></mrow><mrow><mi>p</mi><mi>r</mi></mrow><mrow><mi>e</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>+</mo><msubsup><mrow><mi>γ</mi></mrow><mrow><mi>p</mi><mi>r</mi></mrow><mrow><mi>e</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>≤</mo><mn>6</mn></mrow></semantics></math></inline-formula> if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi></mrow></semantics></math></inline-formula> is a graph with <i>m</i> nodes where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>≡</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>2</mn><mo stretchy="false">(</mo><mi>m</mi><mi>o</mi><mi>d</mi><mo> </mo><mn>8</mn><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> = (<i>m</i>/2) for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></semantics></math></inline-formula>. The challenges and limitations of this parameter of paired equitable and equitable domination depends on the degree of the vertex of the graph. Practical applications are discussed in various fields and illustrated using the studied parameter.https://www.mdpi.com/2075-1680/12/12/1134dominationinflated graphcomplement graphNordhaus–Gaddam inequality |
| spellingShingle | Narayanan Kumaran Annamalai Meenakshi Robert Cep Jayavelu Udaya Prakash Ondrej Mizera Equitable and Paired Equitable Domination in Inflated Graphs and Their Complements Axioms domination inflated graph complement graph Nordhaus–Gaddam inequality |
| title | Equitable and Paired Equitable Domination in Inflated Graphs and Their Complements |
| title_full | Equitable and Paired Equitable Domination in Inflated Graphs and Their Complements |
| title_fullStr | Equitable and Paired Equitable Domination in Inflated Graphs and Their Complements |
| title_full_unstemmed | Equitable and Paired Equitable Domination in Inflated Graphs and Their Complements |
| title_short | Equitable and Paired Equitable Domination in Inflated Graphs and Their Complements |
| title_sort | equitable and paired equitable domination in inflated graphs and their complements |
| topic | domination inflated graph complement graph Nordhaus–Gaddam inequality |
| url | https://www.mdpi.com/2075-1680/12/12/1134 |
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