论文标题
摊子获胜时，在树上的破坏者统治游戏
Maker-Breaker domination game on trees when Staller wins
论文作者
论文摘要
在$ g $上玩的制造商 - 破坏者统治游戏中，Dominator的目标是选择一个主导套装，而Staller的目标是声称某个顶点的封闭社区。我们研究摊子可以赢得比赛的案例。如果Dominator（分别，Staller）开始游戏，则$γ_ {\ rm smb}（g）$（resp。，$γ_ {\ rm smb}'（g）$）表示Staller需要获胜的最小动作数量。对于每个正整数$ k $，树木$ t $带有$γ_ {\ rm smb}'（t）= k $，并且证明了$γ_ {\ rm smb}'$的一般上限。令$ s = s（n_1，\ dots，n_ \ ell）$为从星星获得的细分星，分别用$ \ ell $ edges通过细分其边缘$ n_1-1，\ ldots，n_ \ ell-1 $ times。然后，$γ_ {\ rm smb}'（s）$在所有情况下都确定，除非$ \ ell \ ge 4 $，否则每个$ n_i $均匀。当至少两个奇数$ n_i $ s时，获得最简单的公式。如果$ n_1 $和$ n_2 $是两个最小的数字，则$γ_{\ rm smb}'（s（n_1，\ dots，n_ \ ell））= \ lceil \ log_2（n_1+n_1+n_2+n_2+1）\ rceil $。对于毛毛虫，建立了$γ_ {\ rm smb} $的精确公式和$γ_ {\ rm smb}'$的$。
In the Maker-Breaker domination game played on a graph $G$, Dominator's goal is to select a dominating set and Staller's goal is to claim a closed neighborhood of some vertex. We study the cases when Staller can win the game. If Dominator (resp., Staller) starts the game, then $γ_{\rm SMB}(G)$ (resp., $γ_{\rm SMB}'(G)$) denotes the minimum number of moves Staller needs to win. For every positive integer $k$, trees $T$ with $γ_{\rm SMB}'(T)=k$ are characterized and a general upper bound on $γ_{\rm SMB}'$ is proved. Let $S = S(n_1,\dots, n_\ell)$ be the subdivided star obtained from the star with $\ell$ edges by subdividing its edges $n_1-1, \ldots, n_\ell-1$ times, respectively. Then $γ_{\rm SMB}'(S)$ is determined in all the cases except when $\ell\ge 4$ and each $n_i$ is even. The simplest formula is obtained when there are at least two odd $n_i$s. If $n_1$ and $n_2$ are the two smallest such numbers, then $γ_{\rm SMB}'(S(n_1,\dots, n_\ell))=\lceil \log_2(n_1+n_2+1)\rceil$. For caterpillars, exact formulas for $γ_{\rm SMB}$ and for $γ_{\rm SMB}'$ are established.
