We define a new escape game in graphs that we call Nemesis. The game is played on a graph having a subset of vertices labeled as exits and the goal of one of the two players, called the fugitive, is to reach one of these exit vertices. The second player, i.e. the fugitive adversary, is called the Nemesis. Her goal is to trap the fugitive in a connected component which does not contain any exit. At each round of the game, the fugitive moves from one vertex to an adjacent vertex. Then the Nemesis deletes one edge anywhere in the graph. The game ends when either the fugitive reached an exit or when he is in a connected component that does not contain any exit. In trees and graphs of maximum degree bounded by 3, Nemesis can be solved in linear time. We also show that a variant of the game called Blizzard where only edges adjacent to the position of the fugitive can be deleted also admits a linear time solution. For arbitrary graphs, we show that Nemesis is PSPACE-complete, and that it is NP-hard on planar multigraphs. We extend our results to the related Cat Herding problem, proving its PSPACE-completeness. We also prove that finding a strategy based on a full binary escape tree whose leaves are exists is NP-complete.

翻译：我们定义了一种新的图论逃脱游戏，称为“复仇女神”。该游戏在一个具有标记为出口的顶点子集的图上进行，其中一名玩家（称为逃亡者）的目标是到达其中一个出口顶点。第二名玩家（即逃亡者的对手）被称为“复仇女神”。她的目标是将逃亡者困在不包含任何出口的连通分量中。在游戏的每一轮中，逃亡者从一个顶点移动到相邻顶点。随后，复仇女神可以在图中的任意位置删除一条边。游戏在逃亡者到达出口或处于不包含任何出口的连通分量时结束。在树和最大度不超过3的图中，复仇女神问题可在线性时间内求解。我们还证明了一种称为“暴风雪”的变体游戏（其中仅允许删除与逃亡者位置相邻的边）同样存在线性时间解法。对于任意图，我们证明复仇女神问题是PSPACE完全的，并且在平面多重图上是NP难的。我们将结果扩展到相关的“牧猫问题”，证明了其PSPACE完全性。同时，我们证明了基于以出口为叶节点的完全二叉逃脱树寻找策略的问题是NP完全的。