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Maker–Breaker percolation games I: crossing grids

Part of: Game theory Extremal combinatorics Graph theory

Published online by Cambridge University Press:  15 September 2020

A. Nicholas Day  and
Victor Falgas-Ravry
A. Nicholas Day
Affiliation:
Institutionen för Matematik och Matematisk Statistik, Umeå Universitet, 901 87 Umeå, Sweden.
Victor Falgas-Ravry*
Affiliation:
Institutionen för Matematik och Matematisk Statistik, Umeå Universitet, 901 87 Umeå, Sweden.
*
*Corresponding author. Email: [email protected]
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Abstract

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Motivated by problems in percolation theory, we study the following two-player positional game. Let Λm×n be a rectangular grid-graph with m vertices in each row and n vertices in each column. Two players, Maker and Breaker, play in alternating turns. On each of her turns, Maker claims p (as yet unclaimed) edges of the board Λm×n, while on each of his turns Breaker claims q (as yet unclaimed) edges of the board and destroys them. Maker wins the game if she manages to claim all the edges of a crossing path joining the left-hand side of the board to its right-hand side, otherwise Breaker wins. We call this game the (p, q)-crossing game on Λm×n.

Given m, n ∈ ℕ, for which pairs (p, q) does Maker have a winning strategy for the (p, q)-crossing game on Λm×n? The (1, 1)-case corresponds exactly to the popular game of Bridg-it, which is well understood due to it being a special case of the older Shannon switching game. In this paper we study the general (p, q)-case. Our main result is to establish the following transition.

  • If p ≥ 2q, then Maker wins the game on arbitrarily long versions of the narrowest board possible, that is, Maker has a winning strategy for the (2q, q)-crossing game on Λm×(q+1) for any m ∈ ℕ.

  • If p ≤ 2q − 1, then for every width n of the board, Breaker has a winning strategy for the (p, q)-crossing game on Λm×n for all sufficiently large board-lengths m.

Our winning strategies in both cases adapt more generally to other grids and crossing games. In addition we pose many new questions and problems.

MSC classification

Primary: 05C57: Games on graphs
Secondary: 05D99: None of the above, but in this section 91A43: Games involving graphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 2 , March 2021 , pp. 200 - 227
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020. Published by Cambridge University Press

Footnotes

†

Research supported by Swedish Research Council grant 2016-03488.

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