关于在回避者-执行者游戏中快速获胜
On winning fast in Avoider-Enforcer games
摘要：
我们分析了三个基本的位置游戏的无偏避免者-执行者游戏的持续时间。所有的博弈都是在完全图的$n$顶点上的边上进行的，Avoider的目标分别是保持图是外平面的、无菱形的和$k$ -退化的.很明显，这三场比赛都是执行者的胜利，我们的主要兴趣在于确定避免者在输掉之前可以玩的最大数量的移动。极值图论提供了一个通用的上界的数量Avoider的举动。事实证明，对于所有三个博弈，我们都设法获得了一个下界，这个下界与上界之间的距离只是一个加法常数。特别是，我们展示了一个策略，避免保持他的图outerplanar至少$2n-8$移动，只有6短的最大可能。一个无菱形图最多可以有$d(n)=\lceil\frac{3n-5}{2}\rceil$边，并且证明了Avoider至少可以走$d(n)-3$步.最后，如果$k$是小相比，$n$，我们表明，Avoider可以保持他的图$k$ -退化多达$e(n)$移动，其中$e(n)$是边缘的最大数量$k$ -退化图可以有。
英文摘要：
We analyze the duration of the unbiased Avoider-Enforcer game for three basic positional games. All the games are played on the edges of the complete graph on $n$ vertices, and Avoider's goal is to keep his graph outerplanar, diamond-free and $k$-degenerate, respectively. It is clear that all three games are Enforcer's wins, and our main interest lies in determining the largest number of moves Avoider can play before losing. Extremal graph theory offers a general upper bound for the number of Avoider's moves. As it turns out, for all three games we manage to obtain a lower bound that is just an additive constant away from that upper bound. In particular, we exhibit a strategy for Avoider to keep his graph outerplanar for at least $2n-8$ moves, being just 6 short of the maximum possible. A diamond-free graph can have at most $d(n)=\lceil\frac{3n-5}{2}\rceil$ edges, and we prove that Avoider can play for at least $d(n)-3$ moves. Finally, if $k$ is small compared to $n$, we show that Avoider can keep his graph $k$-degenerate for as many as $e(n)$ moves, where $e(n)$ is the maximum number of edges a $k$-degenerate graph can have.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
