On Roman balanced domination of graphs
Let $ G $ be a graph with vertex set $ V $. A function $ f $ : $ V\to \{-1, 0, 2\} $ is called a Roman balanced dominating function (RBDF) of $ G $ if $ \sum_{u\in N_G[v]}f(u) = 0 $ for each vertex $ v\in V $. The maximum (resp. minimum) Roman balanced domination number $ \gamma^{M}_{Rb}(G) $ (resp....
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AIMS Press
2024-12-01
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20241707 |
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| author | Mingyu Zhang Junxia Zhang |
| author_facet | Mingyu Zhang Junxia Zhang |
| author_sort | Mingyu Zhang |
| collection | DOAJ |
| description | Let $ G $ be a graph with vertex set $ V $. A function $ f $ : $ V\to \{-1, 0, 2\} $ is called a Roman balanced dominating function (RBDF) of $ G $ if $ \sum_{u\in N_G[v]}f(u) = 0 $ for each vertex $ v\in V $. The maximum (resp. minimum) Roman balanced domination number $ \gamma^{M}_{Rb}(G) $ (resp. $ \gamma^{m}_{Rb}(G) $) is the maximum (resp. minimum) value of $ \sum_{v\in V} f(v) $ among all Roman balanced dominating functions $ f $. A graph $ G $ is called $ Rd $-balanced if $ \gamma^{M}_{Rb}(G) = \gamma^{m}_{Rb}(G) = 0 $. In this paper, we obtain several upper and lower bounds on $ \gamma^{M}_{Rb}(G) $ and $ \gamma^{m}_{Rb}(G) $ and further determine several classes of $ Rd $-balanced graphs. |
| format | Article |
| id | doaj-art-9ac54719065940a99f88110e4fa77f08 |
| institution | Kabale University |
| issn | 2473-6988 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj-art-9ac54719065940a99f88110e4fa77f082025-01-23T07:53:25ZengAIMS PressAIMS Mathematics2473-69882024-12-01912360013601110.3934/math.20241707On Roman balanced domination of graphsMingyu Zhang0Junxia Zhang1School of Mathematics and Statistics, Shanxi Datong University, Datong, Shanxi 037009, ChinaSchool of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, ChinaLet $ G $ be a graph with vertex set $ V $. A function $ f $ : $ V\to \{-1, 0, 2\} $ is called a Roman balanced dominating function (RBDF) of $ G $ if $ \sum_{u\in N_G[v]}f(u) = 0 $ for each vertex $ v\in V $. The maximum (resp. minimum) Roman balanced domination number $ \gamma^{M}_{Rb}(G) $ (resp. $ \gamma^{m}_{Rb}(G) $) is the maximum (resp. minimum) value of $ \sum_{v\in V} f(v) $ among all Roman balanced dominating functions $ f $. A graph $ G $ is called $ Rd $-balanced if $ \gamma^{M}_{Rb}(G) = \gamma^{m}_{Rb}(G) = 0 $. In this paper, we obtain several upper and lower bounds on $ \gamma^{M}_{Rb}(G) $ and $ \gamma^{m}_{Rb}(G) $ and further determine several classes of $ Rd $-balanced graphs.https://www.aimspress.com/article/doi/10.3934/math.20241707roman balanced dominating functionroman balanced domination number$ rd $-balanced graph |
| spellingShingle | Mingyu Zhang Junxia Zhang On Roman balanced domination of graphs AIMS Mathematics roman balanced dominating function roman balanced domination number $ rd $-balanced graph |
| title | On Roman balanced domination of graphs |
| title_full | On Roman balanced domination of graphs |
| title_fullStr | On Roman balanced domination of graphs |
| title_full_unstemmed | On Roman balanced domination of graphs |
| title_short | On Roman balanced domination of graphs |
| title_sort | on roman balanced domination of graphs |
| topic | roman balanced dominating function roman balanced domination number $ rd $-balanced graph |
| url | https://www.aimspress.com/article/doi/10.3934/math.20241707 |
| work_keys_str_mv | AT mingyuzhang onromanbalanceddominationofgraphs AT junxiazhang onromanbalanceddominationofgraphs |