无圈竞争游戏
The Enclaveless Competition Game
摘要：
对于图$G$中的顶点子集$S$，如果$v$及其所有邻居都在$S$中，则顶点$v \in S$是$S$的飞地，其中$v$的邻居是与$v$相邻的顶点。如果集合$S$不包含任何安全区，则该集合是无安全区的。$G$的无区域数$Ψ(G)$是$G$中的无区域集合的最大基数。如Slater在1997年首次观察到的[J. Res. Nat. Bur. Standards 82（1977），197- 202]，如果$G$是一个具有$n$顶点的图，则有$γ(G) + Ψ(G) = n$，其中$γ(G)$是$G$的被充分研究的控制数。本文继续研究Phillips和Slater在2001年提出的竞争-无飞地博弈[ Graph Theory Notes N. Y. 41（2001），37- 41]并定义如下。两个参与者轮流构造最大无飞地集合$S$，其中一个参与者Maximizer尝试最大化$|S|$，而另一个参与者Minimizer尝试最小化~ $|S|$。$G$的竞争-无飞地游戏编号$Ψ_g^+(G)$是当最大化器开始游戏并且两个玩家最优地玩时玩的顶点的数目。除其他问题外，我们研究的猜想，如果$G$是一个孤立的自由图的顺序$n$，然后$Ψ_g^+(G) \ge \frac{1}{2}n$。我们证明了这个猜想的正则图和无爪图。
英文摘要：
For a subset $S$ of vertices in a graph $G$, a vertex $v \in S$ is an enclave of $S$ if $v$ and all of its neighbors are in $S$, where a neighbor of $v$ is a vertex adjacent to $v$. A set $S$ is enclaveless if it does not contain any enclaves. The enclaveless number $Ψ(G)$ of $G$ is the maximum cardinality of an enclaveless set in $G$. As first observed in 1997 by Slater [J. Res. Nat. Bur. Standards 82 (1977), 197--202], if $G$ is a graph with $n$ vertices, then $γ(G) + Ψ(G) = n$ where $γ(G)$ is the well-studied domination number of $G$. In this paper, we continue the study of the competition-enclaveless game introduced in 2001 by Phillips and Slater [Graph Theory Notes N. Y. 41 (2001), 37--41] and defined as follows. Two players take turns in constructing a maximal enclaveless set $S$, where one player, Maximizer, tries to maximize $|S|$ and one player, Minimizer, tries to minimize~$|S|$. The competition-enclaveless game number $Ψ_g^+(G)$ of $G$ is the number of vertices played when Maximizer starts the game and both players play optimally. We study among other problems the conjecture that if $G$ is an isolate-free graph of order $n$, then $Ψ_g^+(G) \ge \frac{1}{2}n$. We prove this conjecture for regular graphs and for claw-free graphs.
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