论文标题
总连接统治游戏
Total connected domination game
论文作者
论文摘要
根据标准（总）统治游戏，在图$ G $上的（总）连接的统治游戏是由两个玩家和Staller播放的，其额外要求是，在游戏的每个阶段，所选的顶点会诱导连接的$ G $。如果Dominator启动游戏并且两个玩家都可以发挥最佳作用，则游戏期间选择的顶点数为（总）连接的游戏支配数字（$γ_ {\ rm tcg}（g）$）$γ_ {\ rm cg} $ g $。我们表明，$γ_ {\ rm tcg}（g）\ in \ {γ_{\ rm cg}（g），γ_ {\ rm cg}（g） + 1，γ_ {\ rm cg}（g） tcg}（g）=γ_ {\ rm cg} + i $ for $ i \ in \ {0,1,1,2 \} $。构建了一个大型级别的$ 0 $图形，其中包含所有连接的笛卡尔产品图，并具有最小度量至少$ 2 $的连接的直接产品图。我们表明，没有树是$ 2 $，并且表征$ 1 $树。我们提供一个无限的班级家庭，$ 2 $两部分图。
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 ($γ_{\rm tcg}(G)$) $γ_{\rm cg}(G)$ of $G$. We show that $γ_{\rm tcg}(G)\in \{γ_{\rm cg}(G), γ_{\rm cg}(G) + 1, γ_{\rm cg}(G) + 2\}$, and consequently define $G$ as Class $i$ if $γ_{\rm tcg}(G) = γ_{\rm cg} + i$ 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.
