TL;DR
This paper analyzes the computational complexity of graph Ramsey games, proving PSPACE-completeness for all unrestricted variants and solving some instances, highlighting their intractability and providing insights into strategic gameplay.
Contribution
It establishes the PSPACE-completeness of all unrestricted graph Ramsey avoidance and achievement games and solves specific instances based on symmetric binary Ramsey numbers.
Findings
All unrestricted graph Ramsey games are PSPACE-complete.
Some instances based on symmetric binary Ramsey numbers are ultra-strongly solved.
Evidence suggests most cases based on symmetric binary Ramsey numbers are intractable.
Abstract
We consider combinatorial avoidance and achievement games based on graph Ramsey theory: The players take turns in coloring still uncolored edges of a graph G, each player being assigned a distinct color, choosing one edge per move. In avoidance games, completing a monochromatic subgraph isomorphic to another graph A leads to immediate defeat or is forbidden and the first player that cannot move loses. In the avoidance+ variants, both players are free to choose more than one edge per move. In achievement games, the first player that completes a monochromatic subgraph isomorphic to A wins. Erdos & Selfridge (1973) were the first to identify some tractable subcases of these games, followed by a large number of further studies. We complete these investigations by settling the complexity of all unrestricted cases: We prove that general graph Ramsey avoidance, avoidance+, and achievement…
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Taxonomy
TopicsAdvanced Topology and Set Theory · Limits and Structures in Graph Theory · Computability, Logic, AI Algorithms