Weak signed Roman k-domination in digraphs
Let \(k\geq 1\) be an integer, and let \(D\) be a finite and simple digraph with vertex set \(V(D)\). A weak signed Roman \(k\)-dominating function (WSRkDF) on a digraph \(D\) is a function \(f \colon V(D)\rightarrow \{-1,1,2\}\) satisfying the condition that \(\sum_{x \in N^-[v]}f(x)\geq k\) for ea...
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| Format: | Article |
| Language: | English |
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AGH Univeristy of Science and Technology Press
2024-01-01
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| Series: | Opuscula Mathematica |
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| Online Access: | https://www.opuscula.agh.edu.pl/vol44/2/art/opuscula_math_4414.pdf |
| _version_ | 1827381272218959872 |
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| author | Lutz Volkmann |
| author_facet | Lutz Volkmann |
| author_sort | Lutz Volkmann |
| collection | DOAJ |
| description | Let \(k\geq 1\) be an integer, and let \(D\) be a finite and simple digraph with vertex set \(V(D)\). A weak signed Roman \(k\)-dominating function (WSRkDF) on a digraph \(D\) is a function \(f \colon V(D)\rightarrow \{-1,1,2\}\) satisfying the condition that \(\sum_{x \in N^-[v]}f(x)\geq k\) for each \(v\in V(D)\), where \(N^-[v]\) consists of \(v\) and all vertices of \(D\) from which arcs go into \(v\). The weight of a WSRkDF \(f\) is \(w(f)=\sum_{v\in V(D)}f(v)\). The weak signed Roman \(k\)-domination number \(\gamma_{wsR}^k(D)\) is the minimum weight of a WSRkDF on \(D\). In this paper we initiate the study of the weak signed Roman \(k\)-domination number of digraphs, and we present different bounds on \(\gamma_{wsR}^k(D)\). In addition, we determine the weak signed Roman \(k\)-domination number of some classes of digraphs. Some of our results are extensions of well-known properties of the weak signed Roman domination number \(\gamma_{wsR}(D)=\gamma_{wsR}^1(D)\) and the signed Roman \(k\)-domination number \(\gamma_{sR}^k(D).\) |
| first_indexed | 2024-03-08T13:52:48Z |
| format | Article |
| id | doaj.art-fbea9ae4603849dea666fa19227275ef |
| institution | Directory Open Access Journal |
| issn | 1232-9274 |
| language | English |
| last_indexed | 2024-03-08T13:52:48Z |
| publishDate | 2024-01-01 |
| publisher | AGH Univeristy of Science and Technology Press |
| record_format | Article |
| series | Opuscula Mathematica |
| spelling | doaj.art-fbea9ae4603849dea666fa19227275ef2024-01-15T18:49:38ZengAGH Univeristy of Science and Technology PressOpuscula Mathematica1232-92742024-01-01442285296https://doi.org/10.7494/OpMath.2024.44.2.2854414Weak signed Roman k-domination in digraphsLutz Volkmann0Lehrstuhl II für Mathematik, RWTH Aachen University, 52056 Aachen, GermanyLet \(k\geq 1\) be an integer, and let \(D\) be a finite and simple digraph with vertex set \(V(D)\). A weak signed Roman \(k\)-dominating function (WSRkDF) on a digraph \(D\) is a function \(f \colon V(D)\rightarrow \{-1,1,2\}\) satisfying the condition that \(\sum_{x \in N^-[v]}f(x)\geq k\) for each \(v\in V(D)\), where \(N^-[v]\) consists of \(v\) and all vertices of \(D\) from which arcs go into \(v\). The weight of a WSRkDF \(f\) is \(w(f)=\sum_{v\in V(D)}f(v)\). The weak signed Roman \(k\)-domination number \(\gamma_{wsR}^k(D)\) is the minimum weight of a WSRkDF on \(D\). In this paper we initiate the study of the weak signed Roman \(k\)-domination number of digraphs, and we present different bounds on \(\gamma_{wsR}^k(D)\). In addition, we determine the weak signed Roman \(k\)-domination number of some classes of digraphs. Some of our results are extensions of well-known properties of the weak signed Roman domination number \(\gamma_{wsR}(D)=\gamma_{wsR}^1(D)\) and the signed Roman \(k\)-domination number \(\gamma_{sR}^k(D).\)https://www.opuscula.agh.edu.pl/vol44/2/art/opuscula_math_4414.pdfdigraphweak signed roman \(k\)-dominating functionweak signed roman \(k\)-domination numbersigned roman \(k\)-dominating functionsigned roman \(k\)-domination number |
| spellingShingle | Lutz Volkmann Weak signed Roman k-domination in digraphs Opuscula Mathematica digraph weak signed roman \(k\)-dominating function weak signed roman \(k\)-domination number signed roman \(k\)-dominating function signed roman \(k\)-domination number |
| title | Weak signed Roman k-domination in digraphs |
| title_full | Weak signed Roman k-domination in digraphs |
| title_fullStr | Weak signed Roman k-domination in digraphs |
| title_full_unstemmed | Weak signed Roman k-domination in digraphs |
| title_short | Weak signed Roman k-domination in digraphs |
| title_sort | weak signed roman k domination in digraphs |
| topic | digraph weak signed roman \(k\)-dominating function weak signed roman \(k\)-domination number signed roman \(k\)-dominating function signed roman \(k\)-domination number |
| url | https://www.opuscula.agh.edu.pl/vol44/2/art/opuscula_math_4414.pdf |
| work_keys_str_mv | AT lutzvolkmann weaksignedromankdominationindigraphs |