A note on Fibonacci and Lucas number of domination in path
<p>Let <span class="math"><em>G</em> = (<em>V</em>(<em>G</em>), <em>E</em>(<em>G</em>))</span> be a path of order <span class="math"><em>n</em> ≥ 1</span>. Let <span class=&qu...
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| Format: | Article |
| Language: | English |
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Indonesian Combinatorial Society (InaCombS); Graph Theory and Applications (GTA) Research Centre; University of Newcastle, Australia; Institut Teknologi Bandung (ITB), Indonesia
2018-10-01
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| Series: | Electronic Journal of Graph Theory and Applications |
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| Online Access: | https://www.ejgta.org/index.php/ejgta/article/view/333 |
| _version_ | 1811335836435742720 |
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| author | Leomarich F Casinillo |
| author_facet | Leomarich F Casinillo |
| author_sort | Leomarich F Casinillo |
| collection | DOAJ |
| description | <p>Let <span class="math"><em>G</em> = (<em>V</em>(<em>G</em>), <em>E</em>(<em>G</em>))</span> be a path of order <span class="math"><em>n</em> ≥ 1</span>. Let <span class="math"><em>f</em><sub><em>m</em></sub>(<em>G</em>)</span> be a path with <span class="math"><em>m</em> ≥ 0</span> independent dominating vertices which follows a Fibonacci string of binary numbers where <span class="math">1</span> is the dominating vertex. A set <span class="math"><em>F</em>(<em>G</em>)</span> contains all possible <span class="math"><em>f</em><sub><em>m</em></sub>(<em>G</em>)</span>, <span class="math"><em>m</em> ≥ 0, </span> having the cardinality of the Fibonacci number <span class="math"><em>F</em><sub><em>n</em> + 2</sub></span>. Let <span class="math"><em>F</em><sub><em>d</em></sub>(<em>G</em>)</span> be a set of <span class="math"><em>f</em><sub><em>m</em></sub>(<em>G</em>)</span> where <span class="math"><em>m</em> = <em>i</em>(<em>G</em>)</span> and <span class="math"><em>F</em><sub><em>d</em></sub><sup><em>m</em><em>a</em><em>x</em></sup>(<em>G</em>)</span> be a set of paths with maximum independent dominating vertices. Let <span class="math"><em>l</em><sub><em>m</em></sub>(<em>G</em>)</span> be a path with <span class="math"><em>m</em> ≥ 0</span> independent dominating vertices which follows a Lucas string of binary numbers where <span class="math">1</span> is the dominating vertex. A set <span class="math"><em>L</em>(<em>G</em>)</span> contains all possible <span class="math"><em>l</em><sub><em>m</em></sub>(<em>G</em>)</span>, <span class="math"><em>m</em> ≥ 0</span>, having the cardinality of the Lucas number <span class="math"><em>L</em><sub><em>n</em></sub></span>. Let <span class="math"><em>L</em><sub><em>d</em></sub>(<em>G</em>)</span> be a set of <span class="math"><em>l</em><sub><em>m</em></sub>(<em>G</em>)</span> where <span class="math"><em>m</em> = <em>i</em>(<em>G</em>)</span> and <span class="math"><em>L</em><sub><em>d</em></sub><sup><em>m</em><em>a</em><em>x</em></sup>(<em>G</em>)</span> be a set of paths with maximum independent dominating vertices. This paper determines the number of possible elements in the sets <span class="math"><em>F</em><sub><em>d</em></sub>(<em>G</em>)</span>, <span class="math"><em>L</em><sub><em>d</em></sub>(<em>G</em>), <em>F</em><sub><em>d</em></sub><sup><em>m</em><em>a</em><em>x</em></sup>(<em>G</em>)</span> and <span class="math"><em>L</em><sub><em>d</em></sub><sup><em>m</em><em>a</em><em>x</em></sup>(<em>G</em>)</span> by constructing a combinatorial formula. Furthermore, we examine some properties of <span class="math"><em>F</em>(<em>G</em>)</span> and <span class="math"><em>L</em>(<em>G</em>)</span> and give some important results.</p> |
| first_indexed | 2024-04-13T17:29:43Z |
| format | Article |
| id | doaj.art-91fb78e44a2a48eeaa63fb718b9a861c |
| institution | Directory Open Access Journal |
| issn | 2338-2287 |
| language | English |
| last_indexed | 2024-04-13T17:29:43Z |
| publishDate | 2018-10-01 |
| publisher | Indonesian Combinatorial Society (InaCombS); Graph Theory and Applications (GTA) Research Centre; University of Newcastle, Australia; Institut Teknologi Bandung (ITB), Indonesia |
| record_format | Article |
| series | Electronic Journal of Graph Theory and Applications |
| spelling | doaj.art-91fb78e44a2a48eeaa63fb718b9a861c2022-12-22T02:37:36ZengIndonesian Combinatorial Society (InaCombS); Graph Theory and Applications (GTA) Research Centre; University of Newcastle, Australia; Institut Teknologi Bandung (ITB), IndonesiaElectronic Journal of Graph Theory and Applications2338-22872018-10-016231732510.5614/ejgta.2018.6.2.11128A note on Fibonacci and Lucas number of domination in pathLeomarich F Casinillo0Visayas State University<p>Let <span class="math"><em>G</em> = (<em>V</em>(<em>G</em>), <em>E</em>(<em>G</em>))</span> be a path of order <span class="math"><em>n</em> ≥ 1</span>. Let <span class="math"><em>f</em><sub><em>m</em></sub>(<em>G</em>)</span> be a path with <span class="math"><em>m</em> ≥ 0</span> independent dominating vertices which follows a Fibonacci string of binary numbers where <span class="math">1</span> is the dominating vertex. A set <span class="math"><em>F</em>(<em>G</em>)</span> contains all possible <span class="math"><em>f</em><sub><em>m</em></sub>(<em>G</em>)</span>, <span class="math"><em>m</em> ≥ 0, </span> having the cardinality of the Fibonacci number <span class="math"><em>F</em><sub><em>n</em> + 2</sub></span>. Let <span class="math"><em>F</em><sub><em>d</em></sub>(<em>G</em>)</span> be a set of <span class="math"><em>f</em><sub><em>m</em></sub>(<em>G</em>)</span> where <span class="math"><em>m</em> = <em>i</em>(<em>G</em>)</span> and <span class="math"><em>F</em><sub><em>d</em></sub><sup><em>m</em><em>a</em><em>x</em></sup>(<em>G</em>)</span> be a set of paths with maximum independent dominating vertices. Let <span class="math"><em>l</em><sub><em>m</em></sub>(<em>G</em>)</span> be a path with <span class="math"><em>m</em> ≥ 0</span> independent dominating vertices which follows a Lucas string of binary numbers where <span class="math">1</span> is the dominating vertex. A set <span class="math"><em>L</em>(<em>G</em>)</span> contains all possible <span class="math"><em>l</em><sub><em>m</em></sub>(<em>G</em>)</span>, <span class="math"><em>m</em> ≥ 0</span>, having the cardinality of the Lucas number <span class="math"><em>L</em><sub><em>n</em></sub></span>. Let <span class="math"><em>L</em><sub><em>d</em></sub>(<em>G</em>)</span> be a set of <span class="math"><em>l</em><sub><em>m</em></sub>(<em>G</em>)</span> where <span class="math"><em>m</em> = <em>i</em>(<em>G</em>)</span> and <span class="math"><em>L</em><sub><em>d</em></sub><sup><em>m</em><em>a</em><em>x</em></sup>(<em>G</em>)</span> be a set of paths with maximum independent dominating vertices. This paper determines the number of possible elements in the sets <span class="math"><em>F</em><sub><em>d</em></sub>(<em>G</em>)</span>, <span class="math"><em>L</em><sub><em>d</em></sub>(<em>G</em>), <em>F</em><sub><em>d</em></sub><sup><em>m</em><em>a</em><em>x</em></sup>(<em>G</em>)</span> and <span class="math"><em>L</em><sub><em>d</em></sub><sup><em>m</em><em>a</em><em>x</em></sup>(<em>G</em>)</span> by constructing a combinatorial formula. Furthermore, we examine some properties of <span class="math"><em>F</em>(<em>G</em>)</span> and <span class="math"><em>L</em>(<em>G</em>)</span> and give some important results.</p>https://www.ejgta.org/index.php/ejgta/article/view/333fibonacci numbers, lucas numbers, fibonacci path domination, lucas path domination, independent domination |
| spellingShingle | Leomarich F Casinillo A note on Fibonacci and Lucas number of domination in path Electronic Journal of Graph Theory and Applications fibonacci numbers, lucas numbers, fibonacci path domination, lucas path domination, independent domination |
| title | A note on Fibonacci and Lucas number of domination in path |
| title_full | A note on Fibonacci and Lucas number of domination in path |
| title_fullStr | A note on Fibonacci and Lucas number of domination in path |
| title_full_unstemmed | A note on Fibonacci and Lucas number of domination in path |
| title_short | A note on Fibonacci and Lucas number of domination in path |
| title_sort | note on fibonacci and lucas number of domination in path |
| topic | fibonacci numbers, lucas numbers, fibonacci path domination, lucas path domination, independent domination |
| url | https://www.ejgta.org/index.php/ejgta/article/view/333 |
| work_keys_str_mv | AT leomarichfcasinillo anoteonfibonacciandlucasnumberofdominationinpath AT leomarichfcasinillo noteonfibonacciandlucasnumberofdominationinpath |