Trail Trap: a variant of Partizan Edge Geography

2024-05-08

arXiv.org

AI Summary

This paper studies the combinatorial game *Trail Trap*, where two players alternately move a token along unused edges of an undirected graph; the player unable to move loses. The central problem is determining whether the first or second player has a winning strategy on a given graph. The authors formally define the game for the first time and introduce a combinatorial analysis framework based on recursive strategy decomposition and minimal counterexample arguments. They design a polynomial-time algorithm to decide the winner on trees, prove NP-hardness for simple connected bipartite planar graphs, and fully characterize winning/losing positions for complete bipartite graphs $K_{m,n}$ and grid graphs. Key contributions include: (i) establishing the first formal theoretical model of Trail Trap; (ii) providing a complete solution for trees; (iii) pinning down the computational complexity boundary—tractable on trees, NP-hard on restricted planar bipartite graphs; and (iv) proposing several fundamental open problems concerning broader graph classes and structural characterizations.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Multiagent Systems: Mechanism Design
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We study a two-player game played on undirected graphs called {sc Trail Trap}, which is a variant of a game known as {sc 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 it is NP-hard to determine if Player~2 has a winning strategy on {sc Trail Trap}, even for connected bipartite planar graphs with maximum degree $4$. 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.

Problem

• Analyze winning strategies for Trail Trap on trees
• Prove NP-hardness for Player 2 in bipartite planar graphs
• Characterize winning players for complete bipartite and grid graphs

Innovation

• Algorithm for winning strategy in tree graphs
• NP-hard proof for Player 2 in bipartite graphs
• Winning strategy analysis for complete bipartite graphs