Given a graph $G$, a set $S$ of vertices in $G$ is a general position set if no triple of vertices from $S$ lie on a common shortest path in $G$. The general position achievement/avoidance game is played on a graph $G$ by players A and B who alternately select vertices of $G$. A selection of a vertex by a player is a legal move if it has not been selected before and the set of selected vertices so far forms a general position set of $G$. The player who picks the last vertex is the winner in the general position achievement game and is the loser in the avoidance game. In this paper, we prove that the general position achievement/avoidance games are PSPACE-complete even on graphs with diameter at most 4. For this, we prove that the \textit{mis\`ere} play of the classical Node Kayles game is also PSPACE-complete. As positive results, we obtain linear time algorithms to decide the winning player of the general position avoidance game in rook's graphs, grids, cylinders, and lexicographic products with complete second factors.

翻译：给定图$G$，$G$中顶点集$S$称为通用位置集，若$S$中任意三个顶点不位于$G$的同一最短路径上。通用位置达成/规避游戏在图上进行，玩家A和B轮流选取$G$中的顶点。若某顶点之前未被选取且当前已选顶点集构成$G$的通用位置集，则该选取为合法操作。在通用位置达成游戏中，选取最后一个顶点的玩家获胜；在规避游戏中，该玩家失败。本文证明，即便在直径不超过4的图中，通用位置达成/规避游戏均为PSPACE完全的。为此，我们证明了经典节点凯尔斯游戏的反常玩法也是PSPACE完全的。作为正面结果，我们获得了线性时间算法，用以判定车图、网格图、圆柱图及与完全图作为第二因子的字典积图中的通用位置规避游戏的获胜玩家。