Total connected domination game
The (total) connected domination game on a graph \(G\) is played by two players, Dominator and Staller, according to the standard (total) domination game with the additional requirement that at each stage of the game the selected vertices induce a connected subgraph of \(G\). If Dominator starts the...
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| Format: | Article |
| Language: | English |
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AGH Univeristy of Science and Technology Press
2021-07-01
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| Series: | Opuscula Mathematica |
| Subjects: | |
| Online Access: | https://www.opuscula.agh.edu.pl/vol41/4/art/opuscula_math_4122.pdf |
| _version_ | 1818718132134477824 |
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| author | Csilla Bujtás Michael A. Henning Vesna Iršič Sandi Klavžar |
| author_facet | Csilla Bujtás Michael A. Henning Vesna Iršič Sandi Klavžar |
| author_sort | Csilla Bujtás |
| collection | DOAJ |
| description | The (total) connected domination game on a graph \(G\) is played by two players, Dominator and Staller, according to the standard (total) domination game with the additional requirement that at each stage of the game the selected vertices induce a connected subgraph of \(G\). If Dominator starts the game and both players play optimally, then the number of vertices selected during the game is the (total) connected game domination number (\(\gamma_{\rm tcg}(G)\)) \(\gamma_{\rm cg}(G)\) of \(G\). We show that \(\gamma_{\rm tcg}(G) \in \{\gamma_{\rm cg}(G),\gamma_{\rm cg}(G) + 1,\gamma_{\rm cg}(G) + 2\}\), and consequently define \(G\) as Class \(i\) if \(\gamma_{\rm tcg}(G) = \gamma_{\rm cg} + i\) for \(i \in \{0,1,2\}\). A large family of Class \(0\) graphs is constructed which contains all connected Cartesian product graphs and connected direct product graphs with minumum degree at least \(2\). We show that no tree is Class \(2\) and characterize Class \(1\) trees. We provide an infinite family of Class \(2\) bipartite graphs. |
| first_indexed | 2024-12-17T19:46:11Z |
| format | Article |
| id | doaj.art-6b151fbeaa5f48c4820b2e7111b2bfd1 |
| institution | Directory Open Access Journal |
| issn | 1232-9274 |
| language | English |
| last_indexed | 2024-12-17T19:46:11Z |
| publishDate | 2021-07-01 |
| publisher | AGH Univeristy of Science and Technology Press |
| record_format | Article |
| series | Opuscula Mathematica |
| spelling | doaj.art-6b151fbeaa5f48c4820b2e7111b2bfd12022-12-21T21:34:52ZengAGH Univeristy of Science and Technology PressOpuscula Mathematica1232-92742021-07-01414453464https://doi.org/10.7494/OpMath.2021.41.4.4534122Total connected domination gameCsilla Bujtás0https://orcid.org/0000-0002-0511-5291Michael A. Henning1Vesna Iršič2Sandi Klavžar3University of Ljubljana, Faculty of Mathematics and Physics, Ljubljana, SloveniaUniversity of Johannesburg, Department of Mathematics and Applied Mathematics, Auckland Park, 2006 South AfricaUniversity of Ljubljana, Faculty of Mathematics and Physics, Ljubljana, SloveniaUniversity of Ljubljana, Faculty of Mathematics and Physics, Ljubljana, SloveniaThe (total) connected domination game on a graph \(G\) is played by two players, Dominator and Staller, according to the standard (total) domination game with the additional requirement that at each stage of the game the selected vertices induce a connected subgraph of \(G\). If Dominator starts the game and both players play optimally, then the number of vertices selected during the game is the (total) connected game domination number (\(\gamma_{\rm tcg}(G)\)) \(\gamma_{\rm cg}(G)\) of \(G\). We show that \(\gamma_{\rm tcg}(G) \in \{\gamma_{\rm cg}(G),\gamma_{\rm cg}(G) + 1,\gamma_{\rm cg}(G) + 2\}\), and consequently define \(G\) as Class \(i\) if \(\gamma_{\rm tcg}(G) = \gamma_{\rm cg} + i\) for \(i \in \{0,1,2\}\). A large family of Class \(0\) graphs is constructed which contains all connected Cartesian product graphs and connected direct product graphs with minumum degree at least \(2\). We show that no tree is Class \(2\) and characterize Class \(1\) trees. We provide an infinite family of Class \(2\) bipartite graphs.https://www.opuscula.agh.edu.pl/vol41/4/art/opuscula_math_4122.pdfconnected domination gametotal connected domination gamegraph producttree |
| spellingShingle | Csilla Bujtás Michael A. Henning Vesna Iršič Sandi Klavžar Total connected domination game Opuscula Mathematica connected domination game total connected domination game graph product tree |
| title | Total connected domination game |
| title_full | Total connected domination game |
| title_fullStr | Total connected domination game |
| title_full_unstemmed | Total connected domination game |
| title_short | Total connected domination game |
| title_sort | total connected domination game |
| topic | connected domination game total connected domination game graph product tree |
| url | https://www.opuscula.agh.edu.pl/vol41/4/art/opuscula_math_4122.pdf |
| work_keys_str_mv | AT csillabujtas totalconnecteddominationgame AT michaelahenning totalconnecteddominationgame AT vesnairsic totalconnecteddominationgame AT sandiklavzar totalconnecteddominationgame |