Fast winning strategies in Avoider-Enforcer games

TL;DR

This paper investigates the minimum number of moves required for Enforcer to win certain Avoider-Enforcer games on complete graphs, providing precise estimates for games like non-planarity, connectivity, and non-bipartiteness.

Contribution

It introduces the study of game duration in Avoider-Enforcer games and offers precise bounds for Enforcer's winning time in several known winning scenarios.

Findings

Estimated minimum moves for Enforcer to win in various games

Precise bounds for non-planarity, connectivity, and non-bipartiteness games

Initiation of duration analysis in Avoider-Enforcer games

Abstract

In numerous positional games the identity of the winner is easily determined. In this case one of the more interesting questions is not {\em who} wins but rather {\em how fast} can one win. These type of problems were studied earlier for Maker-Breaker games; here we initiate their study for unbiased Avoider-Enforcer games played on the edge set of the complete graph $K_n$ on $n$ vertices. For several games that are known to be an Enforcer's win, we estimate quite precisely the minimum number of moves Enforcer has to play in order to win. We consider the non-planarity game, the connectivity game and the non-bipartite game.