Weakly connected domination critical graphs
A dominating set \(D \subset V(G)\) is a weakly connected dominating set in \(G\) if the subgraph \(G[D]_w = (N_{G}[D],E_w)\) weakly induced by \(D\) is connected, where \(E_w\) is the set of all edges with at least one vertex in \(D\). The weakly connected domination number \(\gamma_w(G)\) of a gra...
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| Format: | Article |
| Language: | English |
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AGH Univeristy of Science and Technology Press
2008-01-01
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| Series: | Opuscula Mathematica |
| Subjects: | |
| Online Access: | http://www.opuscula.agh.edu.pl/vol28/3/art/opuscula_math_2824.pdf |
| _version_ | 1811310402308407296 |
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| author | Magdalena Lemańska Agnieszka Patyk |
| author_facet | Magdalena Lemańska Agnieszka Patyk |
| author_sort | Magdalena Lemańska |
| collection | DOAJ |
| description | A dominating set \(D \subset V(G)\) is a weakly connected dominating set in \(G\) if the subgraph \(G[D]_w = (N_{G}[D],E_w)\) weakly induced by \(D\) is connected, where \(E_w\) is the set of all edges with at least one vertex in \(D\). The weakly connected domination number \(\gamma_w(G)\) of a graph \(G\) is the minimum cardinality among all weakly connected dominating sets in \(G\). The graph is said to be weakly connected domination critical (\(\gamma_w\)-critical) if for each \(u, v \in V(G)\) with \(v\) not adjacent to \(u\), \(\gamma_w(G + vu) \lt \gamma_w (G)\). Further, \(G\) is \(k\)-\(\gamma_w\)-critical if \(\gamma_w(G) = k\) and for each edge \(e \not\in E(G)\), \(\gamma_w(G + e) \lt k\). In this paper we consider weakly connected domination critical graphs and give some properties of \(3\)-\(\gamma_w\)-critical graphs. |
| first_indexed | 2024-04-13T09:59:19Z |
| format | Article |
| id | doaj.art-1fe12038dfba440797c29997ea6c17b6 |
| institution | Directory Open Access Journal |
| issn | 1232-9274 |
| language | English |
| last_indexed | 2024-04-13T09:59:19Z |
| publishDate | 2008-01-01 |
| publisher | AGH Univeristy of Science and Technology Press |
| record_format | Article |
| series | Opuscula Mathematica |
| spelling | doaj.art-1fe12038dfba440797c29997ea6c17b62022-12-22T02:51:17ZengAGH Univeristy of Science and Technology PressOpuscula Mathematica1232-92742008-01-012833253302824Weakly connected domination critical graphsMagdalena Lemańska0Agnieszka Patyk1Gdańsk University of Technology, Department of Technical Physics and Applied Mathematics, Narutowicza 11/12, 80–952 Gdańsk, PolandGdańsk University of Technology, Department of Technical Physics and Applied Mathematics, Narutowicza 11/12, 80–952 Gdańsk, PolandA dominating set \(D \subset V(G)\) is a weakly connected dominating set in \(G\) if the subgraph \(G[D]_w = (N_{G}[D],E_w)\) weakly induced by \(D\) is connected, where \(E_w\) is the set of all edges with at least one vertex in \(D\). The weakly connected domination number \(\gamma_w(G)\) of a graph \(G\) is the minimum cardinality among all weakly connected dominating sets in \(G\). The graph is said to be weakly connected domination critical (\(\gamma_w\)-critical) if for each \(u, v \in V(G)\) with \(v\) not adjacent to \(u\), \(\gamma_w(G + vu) \lt \gamma_w (G)\). Further, \(G\) is \(k\)-\(\gamma_w\)-critical if \(\gamma_w(G) = k\) and for each edge \(e \not\in E(G)\), \(\gamma_w(G + e) \lt k\). In this paper we consider weakly connected domination critical graphs and give some properties of \(3\)-\(\gamma_w\)-critical graphs.http://www.opuscula.agh.edu.pl/vol28/3/art/opuscula_math_2824.pdfweakly connected domination numbertreecritical graphs |
| spellingShingle | Magdalena Lemańska Agnieszka Patyk Weakly connected domination critical graphs Opuscula Mathematica weakly connected domination number tree critical graphs |
| title | Weakly connected domination critical graphs |
| title_full | Weakly connected domination critical graphs |
| title_fullStr | Weakly connected domination critical graphs |
| title_full_unstemmed | Weakly connected domination critical graphs |
| title_short | Weakly connected domination critical graphs |
| title_sort | weakly connected domination critical graphs |
| topic | weakly connected domination number tree critical graphs |
| url | http://www.opuscula.agh.edu.pl/vol28/3/art/opuscula_math_2824.pdf |
| work_keys_str_mv | AT magdalenalemanska weaklyconnecteddominationcriticalgraphs AT agnieszkapatyk weaklyconnecteddominationcriticalgraphs |