The diameter game

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Abstract
A large class of Positional Games are defined on the complete graph on n vertices. The players, Maker and Breaker, take the edges of the graph in turns, and Maker wins iff his subgraph has a given — usually monotone — property. Here we introduce the d‐diameter game, which means that Maker wins iff the diameter of his subgraph is at most d. We investigate the biased version of the game; i.e., when the players may take more than one, and not necessarily the same number of edges, in a turn. Our main result is that we proved that the 2‐diameter game has the following surprising property: Breaker wins the game in which each player chooses one edge per turn, but Maker wins as long as he is permitted to choose 2 edges in each turn whereas Breaker can choose as many as (1/9)n1/8/(lnn)3/8. In addition, we investigate d‐diameter games for d ≥ 3. The diameter games are strongly related to the degree games. Thus, we also provide a generalization of the fair degree game for the biased case. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009

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