A Symmetric Strategy in Graph Avoidance Games

TL;DR

This paper investigates symmetric strategies in a graph avoidance game where two players color edges to avoid creating a monochromatic forbidden subgraph, exploring graph classes that admit such strategies and their computational aspects.

Contribution

It characterizes graphs that always allow a symmetric strategy for all forbidden subgraphs and examines cases where symmetric strategies are possible for specific F.

Findings

Identifies classes of graphs with universal symmetric strategies.

Provides examples where symmetric strategies exist for particular forbidden graphs.

Discusses complexity issues related to symmetric strategies in graph avoidance games.

Abstract

In the graph avoidance game two players alternatingly color edges of a graph G in red and in blue respectively. The player who first creates a monochromatic subgraph isomorphic to a forbidden graph F loses. A symmetric strategy of the second player ensures that, independently of the first player's strategy, the blue and the red subgraph are isomorphic after every round of the game. We address the class of those graphs G that admit a symmetric strategy for all F and discuss relevant graph-theoretic and complexity issues. We also show examples when, though a symmetric strategy on G generally does not exist, it is still available for a particular F.