# Research Topic: Maker-Breaker games with restrictions

### Working Groups: Chair Discrete Mathematics

### Collaborators (MAT): Dr.Â Dennis Clemens, Fabian Hamann, M. Sc., Yannick Mogge, M. Sc.

### Collaborators (External): Pranshu Gupta, Laurin Kirsch

## Description

Biased Maker-Breaker games are played on some hypergraph \((X,\mathcal{F})\), where \(\mathcal{F}\) denotes the family of winning sets. For some biases \(m,b\), during each round of such a game Maker claims (up to) \(m\) elements of \(X\), followed by Breaker claiming (up to) \(b\) elements. If Maker is able to claim all elements of a winning set, she wins the game, otherwise Breaker is declared the winner.

An interesting question to consider is how these games play out, if we restrict Makerâ€™s (and maybe also Breakerâ€™s) moves. For example, we play on a graph \(G\) and allow Maker to only claim edges incident to previously claimed edges by her. These games are called *Connector-Breaker games* and were first introduced by London and PluhĂˇr [LP2018] under the name *PrimMaker-Breaker games*. Another variant is to have Maker only claim edges according to a walk. This variant, called *Walker-Breaker game*, appeared first in a publication by Espig, Frieze, Krivelevich, and Pegden [EFKP2015]. These restrictions lead to interesting new strategies to be considered for both players to enforce a win.

Additionally one can also add restrictions to the board we play on, for example instead of playing on the complete graph \(K_n\) we will play on random graphs \(G \sim G_{n,p}\) with \(n\) vertices, where each edge is contained in the graph with probability \(p\). Here, a typical goal is to find the threshold probability \(p^*\), where a Makerâ€™s (Connectorâ€™s/Walkerâ€™s) win turns into a Breakerâ€™s win.