Further results on the total Italian domination number of trees
Let $ f:V(G)\rightarrow \{0, 1, 2\} $ be a function defined from a connected graph $ G $. Let $ W_i = \{x\in V(G): f(x) = i\} $ for every $ i\in \{0, 1, 2\} $. The function $ f $ is called a total Italian dominating function on $ G $ if $ \sum_{v\in N(x)}f(v)\geq 2 $ for every vertex $ x\in W_0 $ an...
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AIMS Press
2023-03-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2023540?viewType=HTML |
| _version_ | 1797871485651517440 |
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| author | Abel Cabrera-Martínez Andrea Conchado Peiró Juan Manuel Rueda-Vázquez |
| author_facet | Abel Cabrera-Martínez Andrea Conchado Peiró Juan Manuel Rueda-Vázquez |
| author_sort | Abel Cabrera-Martínez |
| collection | DOAJ |
| description | Let $ f:V(G)\rightarrow \{0, 1, 2\} $ be a function defined from a connected graph $ G $. Let $ W_i = \{x\in V(G): f(x) = i\} $ for every $ i\in \{0, 1, 2\} $. The function $ f $ is called a total Italian dominating function on $ G $ if $ \sum_{v\in N(x)}f(v)\geq 2 $ for every vertex $ x\in W_0 $ and if $ \sum_{v\in N(x)}f(v)\geq 1 $ for every vertex $ x\in W_1\cup W_2 $. The total Italian domination number of $ G $, denoted by $ \gamma_{tI}(G) $, is the minimum weight $ \omega(f) = \sum_{x\in V(G)}f(x) $ among all total Italian dominating functions $ f $ on $ G $. In this paper, we provide new lower and upper bounds on the total Italian domination number of trees. In particular, we show that if $ T $ is a tree of order $ n(T)\geq 2 $, then the following inequality chains are satisfied.
(ⅰ) $ 2\gamma(T)\leq \gamma_{tI}(T)\leq n(T)-\gamma(T)+s(T) $,</p>
<p>(ⅱ) $ \frac{n(T)+\gamma(T)+s(T)-l(T)+1}{2}\leq \gamma_{tI}(T)\leq \frac{n(T)+\gamma(T)+l(T)}{2}, $
where $ \gamma(T) $, $ s(T) $ and $ l(T) $ represent the classical domination number, the number of support vertices and the number of leaves of $ T $, respectively. The upper bounds are derived from results obtained for the double domination number of a tree. |
| first_indexed | 2024-04-10T00:44:39Z |
| format | Article |
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| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-04-10T00:44:39Z |
| publishDate | 2023-03-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-1658ac6328f44e37bfec8d951caf67372023-03-14T01:41:18ZengAIMS PressAIMS Mathematics2473-69882023-03-0185106541066410.3934/math.2023540Further results on the total Italian domination number of treesAbel Cabrera-Martínez 0Andrea Conchado Peiró 1Juan Manuel Rueda-Vázquez 21. Universidad de Córdoba, Departamento de Matemáticas, Campus de Rabanales, Spain2. Universitat Politècnica de València, Centre for Quality and Change Management (CQ), Spain1. Universidad de Córdoba, Departamento de Matemáticas, Campus de Rabanales, SpainLet $ f:V(G)\rightarrow \{0, 1, 2\} $ be a function defined from a connected graph $ G $. Let $ W_i = \{x\in V(G): f(x) = i\} $ for every $ i\in \{0, 1, 2\} $. The function $ f $ is called a total Italian dominating function on $ G $ if $ \sum_{v\in N(x)}f(v)\geq 2 $ for every vertex $ x\in W_0 $ and if $ \sum_{v\in N(x)}f(v)\geq 1 $ for every vertex $ x\in W_1\cup W_2 $. The total Italian domination number of $ G $, denoted by $ \gamma_{tI}(G) $, is the minimum weight $ \omega(f) = \sum_{x\in V(G)}f(x) $ among all total Italian dominating functions $ f $ on $ G $. In this paper, we provide new lower and upper bounds on the total Italian domination number of trees. In particular, we show that if $ T $ is a tree of order $ n(T)\geq 2 $, then the following inequality chains are satisfied. (ⅰ) $ 2\gamma(T)\leq \gamma_{tI}(T)\leq n(T)-\gamma(T)+s(T) $,</p> <p>(ⅱ) $ \frac{n(T)+\gamma(T)+s(T)-l(T)+1}{2}\leq \gamma_{tI}(T)\leq \frac{n(T)+\gamma(T)+l(T)}{2}, $ where $ \gamma(T) $, $ s(T) $ and $ l(T) $ represent the classical domination number, the number of support vertices and the number of leaves of $ T $, respectively. The upper bounds are derived from results obtained for the double domination number of a tree.https://www.aimspress.com/article/doi/10.3934/math.2023540?viewType=HTMLtotal italian domination numberdouble domination numberdomination numbertrees |
| spellingShingle | Abel Cabrera-Martínez Andrea Conchado Peiró Juan Manuel Rueda-Vázquez Further results on the total Italian domination number of trees AIMS Mathematics total italian domination number double domination number domination number trees |
| title | Further results on the total Italian domination number of trees |
| title_full | Further results on the total Italian domination number of trees |
| title_fullStr | Further results on the total Italian domination number of trees |
| title_full_unstemmed | Further results on the total Italian domination number of trees |
| title_short | Further results on the total Italian domination number of trees |
| title_sort | further results on the total italian domination number of trees |
| topic | total italian domination number double domination number domination number trees |
| url | https://www.aimspress.com/article/doi/10.3934/math.2023540?viewType=HTML |
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