Some sufficient conditions for a tree to have its weak Roman domination number be equal to its domination number plus 1
Let $ G = (V, E) $ be a simple graph with vertex set $ V $ and edge set $ E $, and let $ f $ be a function $ f:V\mapsto \{0, 1, 2\} $. A vertex $ u $ with $ f(u) = 0 $ is said to be undefended with respect to $ f $ if it is not adjacent to a vertex with positive weight. The function $ f $ is a weak...
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AIMS Press
2023-05-01
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2023904?viewType=HTML |
| _version_ | 1827932565904818176 |
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| author | Jian Yang Yuefen Chen Zhiqiang Li |
| author_facet | Jian Yang Yuefen Chen Zhiqiang Li |
| author_sort | Jian Yang |
| collection | DOAJ |
| description | Let $ G = (V, E) $ be a simple graph with vertex set $ V $ and edge set $ E $, and let $ f $ be a function $ f:V\mapsto \{0, 1, 2\} $. A vertex $ u $ with $ f(u) = 0 $ is said to be undefended with respect to $ f $ if it is not adjacent to a vertex with positive weight. The function $ f $ is a weak Roman dominating function (WRDF) if each vertex $ u $ with $ f(u) = 0 $ is adjacent to a vertex $ v $ with $ f(v) > 0 $ such that the function $ f_{u}:V\mapsto \{0, 1, 2\} $, defined by $ f_{u}(u) = 1 $, $ f_{u}(v) = f(v)-1 $ and $ f_{u}(w) = f(w) $ if $ w\in V-\{u, v\} $, has no undefended vertex. The weight of $ f $ is $ w(f) = \sum_{v\in V}f(v) $. The weak Roman domination number, denoted $ \gamma_{r}(G) $, is the minimum weight of a WRDF in G. The domination number, denoted $ \gamma(G) $, is the minimum cardinality of a dominating set in $ G $. In this paper, we give some sufficient conditions for a tree to have its weak Roman domination number be equal to its domination number plus 1 ($ \gamma_{r}(T) = \gamma(T)+1 $) by recursion and construction. |
| first_indexed | 2024-03-13T07:11:42Z |
| format | Article |
| id | doaj.art-4250501daad34f9090e4e769b9eac48c |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-03-13T07:11:42Z |
| publishDate | 2023-05-01 |
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| series | AIMS Mathematics |
| spelling | doaj.art-4250501daad34f9090e4e769b9eac48c2023-06-06T01:21:33ZengAIMS PressAIMS Mathematics2473-69882023-05-0188177021771810.3934/math.2023904Some sufficient conditions for a tree to have its weak Roman domination number be equal to its domination number plus 1Jian Yang0Yuefen Chen1Zhiqiang Li21. Department of Public Basic Teaching, Henan College of Transportation, Zhengzhou, China2. Department of Public Basic Courses, Nanjing Vocational University of Industry Technology, Nanjing, China3. School of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou, ChinaLet $ G = (V, E) $ be a simple graph with vertex set $ V $ and edge set $ E $, and let $ f $ be a function $ f:V\mapsto \{0, 1, 2\} $. A vertex $ u $ with $ f(u) = 0 $ is said to be undefended with respect to $ f $ if it is not adjacent to a vertex with positive weight. The function $ f $ is a weak Roman dominating function (WRDF) if each vertex $ u $ with $ f(u) = 0 $ is adjacent to a vertex $ v $ with $ f(v) > 0 $ such that the function $ f_{u}:V\mapsto \{0, 1, 2\} $, defined by $ f_{u}(u) = 1 $, $ f_{u}(v) = f(v)-1 $ and $ f_{u}(w) = f(w) $ if $ w\in V-\{u, v\} $, has no undefended vertex. The weight of $ f $ is $ w(f) = \sum_{v\in V}f(v) $. The weak Roman domination number, denoted $ \gamma_{r}(G) $, is the minimum weight of a WRDF in G. The domination number, denoted $ \gamma(G) $, is the minimum cardinality of a dominating set in $ G $. In this paper, we give some sufficient conditions for a tree to have its weak Roman domination number be equal to its domination number plus 1 ($ \gamma_{r}(T) = \gamma(T)+1 $) by recursion and construction. https://www.aimspress.com/article/doi/10.3934/math.2023904?viewType=HTMLweak roman domination numberdomination numbertreestar |
| spellingShingle | Jian Yang Yuefen Chen Zhiqiang Li Some sufficient conditions for a tree to have its weak Roman domination number be equal to its domination number plus 1 AIMS Mathematics weak roman domination number domination number tree star |
| title | Some sufficient conditions for a tree to have its weak Roman domination number be equal to its domination number plus 1 |
| title_full | Some sufficient conditions for a tree to have its weak Roman domination number be equal to its domination number plus 1 |
| title_fullStr | Some sufficient conditions for a tree to have its weak Roman domination number be equal to its domination number plus 1 |
| title_full_unstemmed | Some sufficient conditions for a tree to have its weak Roman domination number be equal to its domination number plus 1 |
| title_short | Some sufficient conditions for a tree to have its weak Roman domination number be equal to its domination number plus 1 |
| title_sort | some sufficient conditions for a tree to have its weak roman domination number be equal to its domination number plus 1 |
| topic | weak roman domination number domination number tree star |
| url | https://www.aimspress.com/article/doi/10.3934/math.2023904?viewType=HTML |
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