Total Domination Game

2021 ◽  
pp. 47-62
Author(s):  
Boštjan Brešar ◽  
Michael A. Henning ◽  
Sandi Klavžar ◽  
Douglas F. Rall
Keyword(s):  
Total Domination ◽  
Total Domination Game ◽  
Domination Game

Effect of vertex-removal on game total domination numbers

2019 ◽  
Vol 13 (07) ◽  
pp. 2050129
Author(s):  
Karnchana Charoensitthichai ◽  
Chalermpong Worawannotai
Keyword(s):  
Domination Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Connected Graphs ◽  
Isolated Vertex ◽  
Vertex Removal ◽  
Total Domination Game ◽  
Domination Game ◽  
Domination Numbers

The total domination game is played on a graph [Formula: see text] by two players, named Dominator and Staller. They alternately select vertices of [Formula: see text]; each chosen vertex totally dominates its neighbors. In this game, each chosen vertex must totally dominates at least one new vertex not totally dominated before. The game ends when all vertices in [Formula: see text] are totally dominated. Dominator’s goal is to finish the game as soon as possible, and Staller’s goal is to prolong it as much as possible. The game total domination number is the number of chosen vertices when both players play optimally, denoted by [Formula: see text] when Dominator starts the game and denoted by [Formula: see text] when Staller starts the game. In this paper, we show that for any graph [Formula: see text] and a vertex [Formula: see text], where [Formula: see text] has no isolated vertex, we have [Formula: see text] and [Formula: see text]. Moreover, all such differences can be realized by some connected graphs.

scholarly journals Progress towards the total domination game 34-conjecture

2016 ◽  
Vol 339 (11) ◽  
pp. 2620-2627 ◽  
Author(s):  
Michael A. Henning ◽  
Douglas F. Rall
Keyword(s):  
Total Domination ◽  
Total Domination Game ◽  
Domination Game

scholarly journals Maker–Breaker total domination game

2020 ◽  
Vol 282 ◽  
pp. 96-107 ◽  
Author(s):  
Valentin Gledel ◽  
Michael A. Henning ◽  
Vesna Iršič ◽  
Sandi Klavžar
Keyword(s):  
Total Domination ◽  
Total Domination Game ◽  
Domination Game

scholarly journals Transversal Game on Hypergraphs and the $\frac{3}{4}$-Conjecture on the Total Domination Game

2016 ◽  
Vol 30 (3) ◽  
pp. 1830-1847 ◽  
Author(s):  
Csilla Bujtás ◽  
Michael A. Henning ◽  
Zsolt Tuza
Keyword(s):  
Total Domination ◽  
Total Domination Game ◽  
Domination Game

scholarly journals Total connected domination game

2021 ◽  
Vol 41 (4) ◽  
pp. 453-464
Author(s):  
Csilla Bujtás ◽  
Michael A. Henning ◽  
Vesna Iršič ◽  
Sandi Klavžar
Keyword(s):  
Domination Number ◽  
Cartesian Product ◽  
Large Family ◽  
Infinite Family ◽  
Connected Subgraph ◽  
Product Graphs ◽  
Class 1 ◽  
Game Domination Number ◽  
Total Domination Game ◽  
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 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.

On graphs with equal total domination and Grundy total domination numbers

2021 ◽  
Author(s):  
Tanja Dravec ◽  
Marko Jakovac ◽  
Tim Kos ◽  
Tilen Marc
Keyword(s):  
Total Domination ◽  
Domination Numbers

scholarly journals On equality in an upper bound for the restrained and total domination numbers of a graph

2007 ◽  
Vol 307 (22) ◽  
pp. 2845-2852 ◽  
Author(s):  
Peter Dankelmann ◽  
David Day ◽  
Johannes H. Hattingh ◽  
Michael A. Henning ◽  
Lisa R. Markus ◽  
...  
Keyword(s):  
Upper Bound ◽  
Total Domination ◽  
Domination Numbers
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