TY  - JOUR
ID  - Bujtás2021
LB  - Bujtás2021
AU  - Bujtás, Csilla
AU  - Henning, Michael A.
AU  - Iršič, Vesna
AU  - Klavžar, Sandi
TI  - Total connected domination game
ST  - Total connected domination game
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2021
VL  - 41
IS  - 4
SP  - 453
EP  - 464
DO  - https://doi.org/10.7494/OpMath.2021.41.4.453
UR  - https://doi.org/10.7494/OpMath.2021.41.4.453
AB  - 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.
KW  - connected domination game
KW  - total connected domination game
KW  - graph product
KW  - tree
SN  - 1232-9274
ER  -