Nemesis, an Escape Game in Graphs

2026-01-20

AI Summary

This study addresses the problem of whether an evader can reach an exit in a graph-based evasion game despite adversarial interference by a Nemesis that deletes edges each round. The authors formally define the Nemesis game and its variant, Blizzard, and employ tools from graph theory, computational complexity, and reduction techniques to delineate complexity boundaries across different graph classes. Their main contributions include linear-time algorithms for trees and graphs with maximum degree at most three, a proof that the problem is PSPACE-complete on general graphs and NP-hard on planar multigraphs, and the establishment of PSPACE-completeness for the Cat Herding problem. Additionally, they show that constructing strategies based on full binary escape trees is NP-complete.

Technology Category

• Search and Optimization: Adversarial Search
• Game Theory and Economic Paradigms: Adversarial Learning
• Multiagent Systems: Adversarial Agents

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Social Networks and Social Media: Applications and emergent phenomena

Abstract

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.

Problem

• escape game
• graph theory
• computational complexity
• PSPACE-complete
• NP-hard

Innovation

• escape game
• graph algorithms
• computational complexity
• PSPACE-completeness
• linear-time algorithm