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arXiv · 1610.07018

P_3-Games on Chordal Bipartite Graphs

Wing-Kai Hon·Ton Kloks·Fu-Hong Liu·Hsiang-Hsuan Liu·Tao-Ming Wang·Yue-Li Wang
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Abstract

Let G=(V,E) be a connected graph. A set U subseteq V is convex if G[U] is connected and all vertices of V\U have at most one neighbor in U. Let sigma(W) denote the unique smallest convex set that contains W subseteq V. Two players play the following game. Consider a convex set U and call it the `playground.' Initially, U = emptyset. When U=V, the player to move loses the game. Otherwise, that player chooses a vertex x in V\U which is at distance at most two from U. The effect of the move is that the playground U changes into sigma(U cup {x}) and the opponent is presented with this new playground. A graph is chordal bipartite if it is bipartite and has no induced cycle of length more than four. In this paper we show that, when G is chordal bipartite, there is a polynomial-time algorithm that computes the Grundy number of the P_3-game played on G. This implies that there is an efficient algorithm to decide whether the first player has a winning strategy.

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Wing-Kai Hon, Ton Kloks, Fu-Hong Liu, Hsiang-Hsuan Liu, Tao-Ming Wang, Yue-Li Wang. 2016-10-22. P_3-Games on Chordal Bipartite Graphs. https://arxiv.org/abs/1610.07018

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