Total Domination on Some Graph Operators
Let <inline-formula><math display="inline"><semantics><mrow><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></semantics></math></inl...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/3/241 |
| _version_ | 1797407211691966464 |
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| author | José M. Sigarreta |
| author_facet | José M. Sigarreta |
| author_sort | José M. Sigarreta |
| collection | DOAJ |
| description | Let <inline-formula><math display="inline"><semantics><mrow><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></semantics></math></inline-formula> be a graph; a set <inline-formula><math display="inline"><semantics><mrow><mi>D</mi><mo>⊆</mo><mi>V</mi></mrow></semantics></math></inline-formula> is a total dominating set if every vertex <inline-formula><math display="inline"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>V</mi></mrow></semantics></math></inline-formula> has, at least, one neighbor in <i>D</i>. The total domination number <inline-formula><math display="inline"><semantics><mrow><msub><mi>γ</mi><mi>t</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is the minimum cardinality among all total dominating sets. Given an arbitrary graph <i>G</i>, we consider some operators on this graph; <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="monospace">S</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>,</mo><mi mathvariant="monospace">R</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula>, and <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="monospace">Q</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula>, and we give bounds or the exact value of the total domination number of these new graphs using some parameters in the original graph <i>G</i>. |
| first_indexed | 2024-03-09T03:38:02Z |
| format | Article |
| id | doaj.art-ff6bc122d9e74b2893746129ccf189cc |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T03:38:02Z |
| publishDate | 2021-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-ff6bc122d9e74b2893746129ccf189cc2023-12-03T14:45:59ZengMDPI AGMathematics2227-73902021-01-019324110.3390/math9030241Total Domination on Some Graph OperatorsJosé M. Sigarreta0Faculty of Mathematics, Autonomous University of Guerrero, Carlos E. Adame 5, Col. La Garita, C. P. 39350 Acapulco, MexicoLet <inline-formula><math display="inline"><semantics><mrow><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></semantics></math></inline-formula> be a graph; a set <inline-formula><math display="inline"><semantics><mrow><mi>D</mi><mo>⊆</mo><mi>V</mi></mrow></semantics></math></inline-formula> is a total dominating set if every vertex <inline-formula><math display="inline"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>V</mi></mrow></semantics></math></inline-formula> has, at least, one neighbor in <i>D</i>. The total domination number <inline-formula><math display="inline"><semantics><mrow><msub><mi>γ</mi><mi>t</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is the minimum cardinality among all total dominating sets. Given an arbitrary graph <i>G</i>, we consider some operators on this graph; <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="monospace">S</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>,</mo><mi mathvariant="monospace">R</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula>, and <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="monospace">Q</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula>, and we give bounds or the exact value of the total domination number of these new graphs using some parameters in the original graph <i>G</i>.https://www.mdpi.com/2227-7390/9/3/241domination theorytotal dominationgraph operators |
| spellingShingle | José M. Sigarreta Total Domination on Some Graph Operators Mathematics domination theory total domination graph operators |
| title | Total Domination on Some Graph Operators |
| title_full | Total Domination on Some Graph Operators |
| title_fullStr | Total Domination on Some Graph Operators |
| title_full_unstemmed | Total Domination on Some Graph Operators |
| title_short | Total Domination on Some Graph Operators |
| title_sort | total domination on some graph operators |
| topic | domination theory total domination graph operators |
| url | https://www.mdpi.com/2227-7390/9/3/241 |
| work_keys_str_mv | AT josemsigarreta totaldominationonsomegraphoperators |