Structure-biased Maker-Breaker Games

TL;DR

This paper studies a new variant of Maker-Breaker games where Breaker must choose edges from a specific structure, revealing how such constraints affect the game's outcome and thresholds.

Contribution

It introduces a structure-biased variant of Maker-Breaker games and determines the threshold biases for various game types under different structural constraints.

Findings

Threshold biases vary significantly with structure constraints.

Certain structures impose major obstructions for Breaker strategies.

The results suggest new directions for classical Maker-Breaker game analysis.

Abstract

In classical Maker-Breaker games on graphs, Maker and Breaker take turns claiming edges; Maker's goal is to claim all of some structure (e.g., a spanning tree, Hamilton cycle, etc.), while Breaker aims to stop her. The standard question considered is how powerful a Breaker Maker can defeat; i.e., for the $(1:b)$-biased game where Breaker takes $b$ edges per turn, how large can $b$ be for Maker to still have a winning strategy, for various possible goal sets? We introduce a variant of this question in which Breaker is required to choose their multiple edges as the edges of (a subgraph of) a given structure (e.g., a matching, clique, etc.) on each turn. We establish the order of magnitude of the threshold biases for triangle games, connectivity games, and Hamiltonicity games under clique, matching, and star biases respectively. We conclude that in many cases structure imposes major obstruction to Breaker, opening up a set of games whose strategies deviate from the classical biased Maker-Breaker game strategies, and shedding light on the types of Breaker strategies that may or may not work to prove tighter bounds in the classical setting.