Efficient Winning Strategies in Random‐Turn Maker–Breaker Games

Ferber, Asaf; Krivelevich, Michael; Kronenberg, Gal

doi:10.1002/jgt.22072

Cited by 4 publications

(2 citation statements)

References 23 publications

(53 reference statements)

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“…Maker-Breaker games (as well as other positional games) have been introduced by Erdős and Selfridge in [13], and since then have been the subject of numerous studies, see [2,3,14,15]. Maker-Breaker games are played on hypergraphs by two players called Maker and Breaker.…”

Section: Introductionmentioning

confidence: 99%

Maker–Breaker Domination Number

Gledel¹,

Iršič²,

Klavžar³

2019

Bull. Malays. Math. Sci. Soc.

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The Maker-Breaker domination game is played on a graph G by Dominator and Staller. The players alternatively select a vertex of G that was not yet chosen in the course of the game. Dominator wins if at some point the vertices he has chosen form a dominating set. Staller wins if Dominator cannot form a dominating set. In this paper we introduce the Maker-Breaker domination number γ MB (G) of G as the minimum number of moves of Dominator to win the game provided that he has a winning strategy and is the first to play. If Staller plays first, then the corresponding invariant is denoted γ ′ MB (G). Comparing the two invariants it turns out that they behave much differently than the related game domination numbers. The invariant γ MB (G) is also compared with the domination number. Using the Erdős-Selfridge Criterion a large class of graphs G is found for which γ MB (G) > γ(G) holds. Residual graphs are introduced and used to bound/determine γ MB (G) and γ ′ MB (G). Using residual graphs, γ MB (T ) and γ ′ MB (T ) are determined for an arbitrary tree. The invariants are also obtained for cycles and bounded for union of graphs. A list of open problems and directions for further investigations is given.

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Section: Introductionmentioning

confidence: 99%

Maker–Breaker Domination Number

Gledel¹,

Iršič²,

Klavžar³

2019

Bull. Malays. Math. Sci. Soc.

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“…A p-random-turn Maker-Breaker game is the same as an ordinary Maker-Breaker game, except that instead of alternating turns, before each turn a biased coin is being tossed and Maker plays this turn with probability p independently of all other turns. Maker-Breaker games under this setting were considered in [18] and in [10].…”

Section: Introductionmentioning

confidence: 99%

Random-Player Maker-Breaker games

Krivelevich

Kronenberg

2015

Electron. J. Combin.

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In a (1 : b) Maker-Breaker game, one of the central questions is to find the maximal value of b that allows Maker to win the game (that is, the critical bias b * ). Erdős conjectured that the critical bias for many Maker-Breaker games played on the edge set of K n is the same as if both players claim edges randomly. Indeed, in many Maker-Breaker games, "Erdős Paradigm" turned out to be true. Therefore, the next natural question to ask is the (typical) value of the critical bias for Maker-Breaker games where only one player claims edges randomly. A random-player Maker-Breaker game is a two-player game, played the same as an ordinary (biased) Maker-Breaker game, except that one player plays according to his best strategy and claims one element in each round, while the other plays randomly and claims b (or m) elements. In fact, for every (ordinary) Maker-Breaker game, there are two different random-player versions; the (1 : b) random-Breaker game and the (m : 1) random-Maker game. We analyze the random-player version of several classical Maker-Breaker games such as the Hamilton cycle game, the perfect-matching game and the k-vertex-connectivity game (played on the edge set of K n ). For each of these games we find or estimate the asymptotic values of the bias (either b or m) that allow each player to typically win the game. In fact, we provide the "smart" player with an explicit winning strategy for the corresponding value of the bias.

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