Bounds on the Signed Roman k-Domination Number of a Digraph
Let k be a positive integer. A signed Roman k-dominating function (SRkDF) on a digraph D is a function f : V (D) → {−1, 1, 2} satisfying the conditions that (i) Σx∈N−[v]f(x) ≥ k for each v ∈ V (D), where N−[v] is the closed in-neighborhood of v, and (ii) each vertex u for which f(u) = −1 has an in-n...
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| Format: | Article |
| Language: | English |
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University of Zielona Góra
2019-02-01
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| Series: | Discussiones Mathematicae Graph Theory |
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| Online Access: | https://doi.org/10.7151/dmgt.2068 |
| _version_ | 1797713122169978880 |
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| author | Chen Xiaodan Hao Guoliang Volkmann Lutz |
| author_facet | Chen Xiaodan Hao Guoliang Volkmann Lutz |
| author_sort | Chen Xiaodan |
| collection | DOAJ |
| description | Let k be a positive integer. A signed Roman k-dominating function (SRkDF) on a digraph D is a function f : V (D) → {−1, 1, 2} satisfying the conditions that (i) Σx∈N−[v]f(x) ≥ k for each v ∈ V (D), where N−[v] is the closed in-neighborhood of v, and (ii) each vertex u for which f(u) = −1 has an in-neighbor v for which f(v) = 2. The weight of an SRkDF f is Σv∈V (D)f(v). The signed Roman k-domination number γksR(D) of a digraph D is the minimum weight of an SRkDF on D. We determine the exact values of the signed Roman k-domination number of some special classes of digraphs and establish some bounds on the signed Roman k-domination number of general digraphs. In particular, for an oriented tree T of order n, we show that γ2sR(T) ≥ (n + 3)/2, and we characterize the oriented trees achieving this lower bound. |
| first_indexed | 2024-03-12T07:32:23Z |
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| id | doaj.art-0c70d0535d794c4fa357d9a2506a6de3 |
| institution | Directory Open Access Journal |
| issn | 2083-5892 |
| language | English |
| last_indexed | 2024-03-12T07:32:23Z |
| publishDate | 2019-02-01 |
| publisher | University of Zielona Góra |
| record_format | Article |
| series | Discussiones Mathematicae Graph Theory |
| spelling | doaj.art-0c70d0535d794c4fa357d9a2506a6de32023-09-02T21:43:04ZengUniversity of Zielona GóraDiscussiones Mathematicae Graph Theory2083-58922019-02-01391677910.7151/dmgt.2068dmgt.2068Bounds on the Signed Roman k-Domination Number of a DigraphChen Xiaodan0Hao Guoliang1Volkmann Lutz2College of Mathematics and Information Science Guangxi University Nanning, P. R.Nanning, ChinaCollege of Science East China University of Technology Nanchang, P.R.Nanchang, ChinaLehrstuhl II für Mathematik, RWTH Aachen University,Aachen, GermanyLet k be a positive integer. A signed Roman k-dominating function (SRkDF) on a digraph D is a function f : V (D) → {−1, 1, 2} satisfying the conditions that (i) Σx∈N−[v]f(x) ≥ k for each v ∈ V (D), where N−[v] is the closed in-neighborhood of v, and (ii) each vertex u for which f(u) = −1 has an in-neighbor v for which f(v) = 2. The weight of an SRkDF f is Σv∈V (D)f(v). The signed Roman k-domination number γksR(D) of a digraph D is the minimum weight of an SRkDF on D. We determine the exact values of the signed Roman k-domination number of some special classes of digraphs and establish some bounds on the signed Roman k-domination number of general digraphs. In particular, for an oriented tree T of order n, we show that γ2sR(T) ≥ (n + 3)/2, and we characterize the oriented trees achieving this lower bound.https://doi.org/10.7151/dmgt.2068signed roman k-dominating functionsigned roman k-domination numberdigraphoriented tree05c6905c20 |
| spellingShingle | Chen Xiaodan Hao Guoliang Volkmann Lutz Bounds on the Signed Roman k-Domination Number of a Digraph Discussiones Mathematicae Graph Theory signed roman k-dominating function signed roman k-domination number digraph oriented tree 05c69 05c20 |
| title | Bounds on the Signed Roman k-Domination Number of a Digraph |
| title_full | Bounds on the Signed Roman k-Domination Number of a Digraph |
| title_fullStr | Bounds on the Signed Roman k-Domination Number of a Digraph |
| title_full_unstemmed | Bounds on the Signed Roman k-Domination Number of a Digraph |
| title_short | Bounds on the Signed Roman k-Domination Number of a Digraph |
| title_sort | bounds on the signed roman k domination number of a digraph |
| topic | signed roman k-dominating function signed roman k-domination number digraph oriented tree 05c69 05c20 |
| url | https://doi.org/10.7151/dmgt.2068 |
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