We introduce a two-player game played on undirected graphs called Trail Trap, which is a variant of a game known as Partizan Edge Geography. One player starts by choosing any edge and moving a token from one endpoint to the other; the other player then chooses a different edge and does the same. Alternating turns, each player moves their token along an unused edge from its current vertex to an adjacent vertex, until one player cannot move and loses. We present an algorithm to determine which player has a winning strategy when the graph is a tree, and partially characterize the trees on which a given player wins. Additionally, we show that Trail Trap is NP-hard, even for connected bipartite planar graphs with maximum degree $4$ as well as for disconnected graphs. We determine which player has a winning strategy for certain subclasses of complete bipartite graphs and grid graphs, and we propose several open problems for further study.

翻译：我们提出一种在无向图上进行双人游戏，称为径迹陷阱，它是名为党派边缘地理游戏的变体。一名玩家先选择任意一条边，将标记从一个端点移至另一个端点；另一名玩家随后选择一条不同的边，并执行相同操作。双方轮流进行，每位玩家沿着未使用过的边，从当前顶点移动其标记到相邻顶点，直到一方无法移动而输掉游戏。我们提出一种算法来确定当图为树时哪位玩家有获胜策略，并部分刻画了给定玩家获胜的树结构。此外，我们证明径迹陷阱是NP困难的，即使对于最大度为4的连通二分平面图以及不连通图也是如此。我们确定了完全二分图和网格图特定子类中哪位玩家有获胜策略，并提出了若干供进一步研究的开放问题。