# The random strategy in Maker-Breaker graph minor games

## TL;DR

This paper demonstrates that in Maker-Breaker graph minor games, the random strategy for Maker is nearly optimal under certain conditions, extending previous results from subgraph games to minors.

## Contribution

It proves that the random strategy is asymptotically optimal in the $H$-graph minor game, generalizing prior work on subgraph games.

## Key findings

- Random strategy is essentially optimal for certain $H$ in the minor game.
- The result extends the understanding of optimal strategies from subgraph to minor games.
- Conditions under which the random strategy is within a factor of $1+o(1)$]

## Abstract

In a $(1:b)$ biased Maker-Breaker game, how good a strategy is for a player can be measured by the bias range for which its rival can win, choosing an appropriate counterstrategy. Bednarska and {\L}uczak proved that, in the $H$-subgraph game, the uniformly random strategy for Maker is essentially optimal with high probability. Here we prove an analogous result for the $H$-graph minor game, and we study for which choices of $H$ the random strategy is within a factor of $1+o(1)$ of being optimal.