AI 中文总结
在连通图上引入竞争性两人零强迫博弈，玩家交替操作以最小化自身种子数，定义最优玩法并证明相关不等式，确定多种图的该博弈参数值，刻画特定参数的图类，还研究其非子图单调性并验证参数上界。
AI 中文摘要
我们在连通图上引入了一种竞争性两人零强迫博弈。Alice和Bob交替为白色顶点播种或执行自身颜色的合法零强迫操作，每位玩家力求最小化自身的种子数量。力保持规则可防止对手已建立的力被不必要地阻碍。由于当前玩家的不同后续行动可能效果相当，最优玩法由集值逆向归纳定义，而\textit{Z}\textsubscript{g}(\textit{G})是由此产生的最优结果中种子的最小总数。我们证明\textit{Z}\textsubscript{g}(\textit{G})≥\textit{Z}(\textit{G})，确定路径、环、星型图、完全图和完全二部图的\textit{Z}\textsubscript{g}值，并通过交替两链强迫 schedule 刻画\textit{Z}\textsubscript{g}(\textit{G})=2的图。我们还表明\textit{Z}\textsubscript{g}不是子图单调的，边细分可增大或减小该参数。通过9阶以内的精确计算验证\textit{Z}\textsubscript{g}(\textit{G})≤2\textit{Z}(\textit{G})。
英文摘要
We introduce a competitive two-player zero forcing game on a connected graph. Alice and Bob alternately seed white vertices or perform legal zero forces in their own colours, and each player seeks to minimise their own number of seeds. A force preservation rule prevents avoidable blocking of an opponent's established force. Because distinct continuations can be equally good for the player to move, optimal play is defined by a set-valued backward induction, and \(Z_g(G)\) is the minimum total number of seeds among the resulting optimal outcomes. We prove that \(Z_g(G)\geq Z(G)\), determine \(Z_g\) for paths, cycles, stars, complete graphs, and complete bipartite graphs, and characterise the graphs with \(Z_g(G)=2\) by an alternating two-chain forcing schedule. We also show that \(Z_g\) is not minor-monotone and that edge subdivision can either increase or decrease the parameter. Exact computation verifies \(Z_g(G)\leq2Z(G)\) through order nine.
