Nemesis, an Escape Game in Graphs

TL;DR

Nemesis presents a novel edge-deletion pursuit game on graphs where a fugitive seeks exits while an adversary progressively removes edges. The authors develop a tight dichotomy: polynomial-time solvable instances on trees and graphs of maximum degree 3 (via the binary escape tree criterion) and linear-time solvable Blizzard, contrasted with PSPACE-complete hardness for arbitrary multigraphs and graphs (via reductions from QSAT and PMR3SAT). A key conceptual tool is the binary escape tree, enabling linear-time certification of winning strategies in tractable cases, while the reductions establish broad hardness, linking Nemesis to Cat Herding. The work thus maps the computational landscape of Nemesis across graph classes and connects it to broader pursuit-evasion problems, with implications for related edge-deletion games and combinatorial game analysis.

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.

Figures (8)

• Figure 1: Two games of Nemesis. Exits are red vertices. At each round, the fugitive whose position is represented by the black vertex moves from one edge while the Nemesis removes one edge.
• Figure 2: Eight instances of Nemesis. Can the fugitive, starting from the black vertex, escape?
• Figure 3: Simplification of an input graph. From a) to b), exits are duplicated. From b) to e), regular vertices of degree $1$ or $2$ are pruned. In f), a binary escape tree.
• Figure 4: Nemesis on a square grid. In a), the input graph with the starting vertex in orange and exits in red. In b), with a starting vertex at distance $5$ from the closest exit, the fugitive wins by moving around the grid. In c), with a starting vertex at distance $6$ from the closest exit, the Nemesis wins by cutting one exit at each corner.
• Figure 8: Reduction of a PMR3SAT instance to a Nemesis instance. We reduce the instance of PMR3SAT represented above. All the edges of the Nemesis instance are unbreakable except the edges adjacent to exits (in green). Their mulitplicity is $L+1$ for the clause exists and it is $12K-5+1$ for the main exit.

Theorems & Definitions (15)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Corollary 6
• Remark 7
• Remark 8
• Theorem 9
• Definition 10