Positional Games

Hefetz, Dan; Krivelevich, Michael; Stojaković, Miloš; Szabó, T.

doi:10.1007/978-3-0348-0825-5

Cited by 75 publications

(94 citation statements)

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“…We denote this game by Box(n × s; m : b). Proof The proof is a straightforward adaptation of the argument in [12] (see chapter 3.4.1) where the case b = 1 is handled. At any point during the game we say that a box a is active if it was not previously touched by BoxBreaker.…”

Section: The Game Boxmentioning

confidence: 99%

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Efficient Winning Strategies in Random‐Turn Maker–Breaker Games

Ferber

Krivelevich

Kronenberg

2016

Journal of Graph Theory

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We consider random-turn positional games, introduced by Peres, Schramm, Sheffield and Wilson in 2007. A p-random-turn positional game is a two-player game, played the same as an ordinary positional game, except that instead of alternating turns, a coin is being tossed before each turn to decide the identity of the next player to move (the probability of Player I to move is p). We analyze the random-turn version of several classical MakerBreaker games such as the game Box (introduced by Chvátal and Erdős in 1987), the Hamilton cycle game and the k-vertex-connectivity game (both played on the edge set of K n ). For each of these games we provide each of the players with a (randomized) efficient strategy which typically ensures his win in the asymptotic order of the minimum value of p for which he typically wins the game, assuming optimal strategies of both players.

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Section: The Game Boxmentioning

confidence: 99%

“…In a relevant development, the second author of this paper proved in [15] that the critical bias for the Hamiltonicity game is asymptotically equal to n ln n as well. We refer the reader to [2,12] for more background on positional games in general and on Maker-Breaker games in particular.…”

Section: Introductionmentioning

confidence: 99%

Efficient Winning Strategies in Random‐Turn Maker–Breaker Games

Ferber

Krivelevich

Kronenberg

2016

Journal of Graph Theory

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“…We define a Maker-Breaker game regarding the packability of the input graph G. Maker-Breaker games are widely studied in combinatorial game theory (see [1,9] for a general overview). Two players, M (Maker) and B (Breaker), alternately pack the vertices of G into the host graph, packing one vertex at a time.…”

Section: Maker-breaker Games and Well-packable Graphsmentioning

confidence: 99%

New models of graph-bin packing

Bujtás

Dósa

Imreh

et al. 2016

Theoretical Computer Science

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In [Int. J. Found. Computer Sci. 22 (2011) 1971-1993] the authors introduced a very general problem called Graph-Bin Packing (GBP). It requires a mapping µ : V (G) → V (H) from the vertex set of an input graph G into a fixed host graph H, which, among other conditions, satisfies that for each pair u, v of adjacent vertices the distance of µ(u) and µ(v) in H is between two prescribed bounds. In this paper we propose two online versions of the Graph-Bin packing problem. In both cases the vertices can arrive in an arbitrary order where each new vertex is adjacent to some of the previous ones. One version is a Maker-Breaker game whose rules are defined by the packing conditions. A subclass of Maker-win input graphs is what we call 'well-packable'; it means that a packing of G is obtained whenever the mapping µ(u) is generated by selecting an arbitrary feasible vertex of the host graph for the next vertex of G in each step. The other model is connected-online packing where we are looking for an online algorithm which can always find a feasible packing. In both models we present some sufficient and some necessary conditions for packability. In the connected-online version we also give bounds on the size of used part of the host graph.

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“…Maker wins if at some point of the game he has selected all vertices from one of the hyperedges, while Breaker wins if she can keep him from doing it. See [1] and [16] for general introductions on this field.…”

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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Positional Games

Cited by 75 publications

References 0 publications

Efficient Winning Strategies in Random‐Turn Maker–Breaker Games

Efficient Winning Strategies in Random‐Turn Maker–Breaker Games

New models of graph-bin packing

Maker–Breaker Domination Number

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