Study on a Strong and Weak <i>n</i>-Connected Total Perfect <i>k</i>-Dominating set in Fuzzy Graphs
In this paper, the concept of a strong <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>n</mi></semantics></math></inline-formula>-Connected Total Perfect <inline-formula><mat...
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MDPI AG
2022-09-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/17/3178 |
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| author | Krishnasamy Elavarasan Tharmalingam Gunasekar Lenka Cepova Robert Cep |
| author_facet | Krishnasamy Elavarasan Tharmalingam Gunasekar Lenka Cepova Robert Cep |
| author_sort | Krishnasamy Elavarasan |
| collection | DOAJ |
| description | In this paper, the concept of a strong <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>n</mi></semantics></math></inline-formula>-Connected Total Perfect <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>k</mi></semantics></math></inline-formula>-connected total perfect <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>k</mi></semantics></math></inline-formula>-dominating set and a weak <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>n</mi></semantics></math></inline-formula>-connected total perfect <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>k</mi></semantics></math></inline-formula>-dominating set in fuzzy graphs is introduced. In the current work, the triple-connected total perfect dominating set is modified to an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>n</mi></semantics></math></inline-formula>-connected total perfect <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>k</mi></semantics></math></inline-formula>-dominating set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>n</mi><mrow><mi>c</mi><mi>t</mi><mi>p</mi><mi>k</mi><mi>D</mi></mrow></msub></mrow></semantics></math></inline-formula>(G) and number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>γ</mi><msub><mi>n</mi><mrow><mi>c</mi><mi>t</mi><mi>p</mi><mi>D</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> New definitions are compared with old ones. Strong and weak <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>n</mi></semantics></math></inline-formula>-connected total perfect <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>k</mi></semantics></math></inline-formula>-dominating set and number of fuzzy graphs are obtained. The results of those fuzzy sets are discussed with the definitions of spanning fuzzy graphs, strong and weak arcs, dominating sets, perfect dominating sets, generalization of triple-connected total perfect dominating sets of fuzzy graphs, complete, connected, bipartite, cut node, tree, bridge and some other new notions of fuzzy graphs which are analyzed with a strong and weak <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>n</mi><mrow><mi>c</mi><mi>t</mi><mi>p</mi><mi>k</mi><mi>D</mi></mrow></msub></mrow></semantics></math></inline-formula>(G) set of fuzzy graphs. The order and size of the strong and weak <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>n</mi><mrow><mi>c</mi><mi>t</mi><mi>p</mi><mi>k</mi><mi>D</mi></mrow></msub><mrow><mo>(</mo><mi mathvariant="normal">G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> fuzzy set are studied. Additionally, a few related theorems and statements are analyzed. |
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| institution | Directory Open Access Journal |
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| language | English |
| last_indexed | 2024-03-10T01:32:20Z |
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| spelling | doaj.art-2ed10603c59e4f1fb6d9620b26317b0e2023-11-23T13:39:55ZengMDPI AGMathematics2227-73902022-09-011017317810.3390/math10173178Study on a Strong and Weak <i>n</i>-Connected Total Perfect <i>k</i>-Dominating set in Fuzzy GraphsKrishnasamy Elavarasan0Tharmalingam Gunasekar1Lenka Cepova2Robert Cep3Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai 600 062, IndiaDepartment of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai 600 062, IndiaDepartment of Machining, Assembly and Engineering Metrology, Faculty of Mechanical Engineering, VSB-Technical University of Ostrava, 708 00 Ostrava, Czech RepublicDepartment of Machining, Assembly and Engineering Metrology, Faculty of Mechanical Engineering, VSB-Technical University of Ostrava, 708 00 Ostrava, Czech RepublicIn this paper, the concept of a strong <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>n</mi></semantics></math></inline-formula>-Connected Total Perfect <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>k</mi></semantics></math></inline-formula>-connected total perfect <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>k</mi></semantics></math></inline-formula>-dominating set and a weak <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>n</mi></semantics></math></inline-formula>-connected total perfect <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>k</mi></semantics></math></inline-formula>-dominating set in fuzzy graphs is introduced. In the current work, the triple-connected total perfect dominating set is modified to an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>n</mi></semantics></math></inline-formula>-connected total perfect <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>k</mi></semantics></math></inline-formula>-dominating set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>n</mi><mrow><mi>c</mi><mi>t</mi><mi>p</mi><mi>k</mi><mi>D</mi></mrow></msub></mrow></semantics></math></inline-formula>(G) and number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>γ</mi><msub><mi>n</mi><mrow><mi>c</mi><mi>t</mi><mi>p</mi><mi>D</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> New definitions are compared with old ones. Strong and weak <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>n</mi></semantics></math></inline-formula>-connected total perfect <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>k</mi></semantics></math></inline-formula>-dominating set and number of fuzzy graphs are obtained. The results of those fuzzy sets are discussed with the definitions of spanning fuzzy graphs, strong and weak arcs, dominating sets, perfect dominating sets, generalization of triple-connected total perfect dominating sets of fuzzy graphs, complete, connected, bipartite, cut node, tree, bridge and some other new notions of fuzzy graphs which are analyzed with a strong and weak <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>n</mi><mrow><mi>c</mi><mi>t</mi><mi>p</mi><mi>k</mi><mi>D</mi></mrow></msub></mrow></semantics></math></inline-formula>(G) set of fuzzy graphs. The order and size of the strong and weak <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>n</mi><mrow><mi>c</mi><mi>t</mi><mi>p</mi><mi>k</mi><mi>D</mi></mrow></msub><mrow><mo>(</mo><mi mathvariant="normal">G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> fuzzy set are studied. Additionally, a few related theorems and statements are analyzed.https://www.mdpi.com/2227-7390/10/17/3178total perfect <i>k</i>-dominating set<i>n</i>-connected total perfect <i>k</i>-dominating setstrong and weak <i>n</i>-connected total perfect <i>k</i>-dominating set and number |
| spellingShingle | Krishnasamy Elavarasan Tharmalingam Gunasekar Lenka Cepova Robert Cep Study on a Strong and Weak <i>n</i>-Connected Total Perfect <i>k</i>-Dominating set in Fuzzy Graphs Mathematics total perfect <i>k</i>-dominating set <i>n</i>-connected total perfect <i>k</i>-dominating set strong and weak <i>n</i>-connected total perfect <i>k</i>-dominating set and number |
| title | Study on a Strong and Weak <i>n</i>-Connected Total Perfect <i>k</i>-Dominating set in Fuzzy Graphs |
| title_full | Study on a Strong and Weak <i>n</i>-Connected Total Perfect <i>k</i>-Dominating set in Fuzzy Graphs |
| title_fullStr | Study on a Strong and Weak <i>n</i>-Connected Total Perfect <i>k</i>-Dominating set in Fuzzy Graphs |
| title_full_unstemmed | Study on a Strong and Weak <i>n</i>-Connected Total Perfect <i>k</i>-Dominating set in Fuzzy Graphs |
| title_short | Study on a Strong and Weak <i>n</i>-Connected Total Perfect <i>k</i>-Dominating set in Fuzzy Graphs |
| title_sort | study on a strong and weak i n i connected total perfect i k i dominating set in fuzzy graphs |
| topic | total perfect <i>k</i>-dominating set <i>n</i>-connected total perfect <i>k</i>-dominating set strong and weak <i>n</i>-connected total perfect <i>k</i>-dominating set and number |
| url | https://www.mdpi.com/2227-7390/10/17/3178 |
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