Geodesic
The Voronoi game on graphs and its complexity
Journal of Graph Algorithms and Applications, Tome 15 (2011) no. 4, pp. 485-501.

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The Voronoi game is a two-person game which is a model for a competitive facility location. The game is played on a continuous domain, and only two special cases (one-dimensional case and one-round case) are well investigated. We introduce the discrete Voronoi game in which the game arena is given as a graph. We first analyze the game when the arena is a large complete k-ary tree, and give an optimal strategy. When both players play optimally, the first player wins when k is odd, and the game ends in a tie for even k. Next we show that the discrete Voronoi game is intractable in general. Even for the one-round case in which the strategy adopted by the first player consist of a fixed single node, deciding whether the second player can win is NP-complete. We also show that deciding whether the second player can win is PSPACE-complete in general.
DOI : 10.7155/jgaa.00235
Keywords: k-trees, NP-completeness, PSPACE-completeness, Voronoi Game 
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Sachio Teramoto; Erik Demaine; Ryuhei Uehara. The Voronoi game on graphs and its complexity. Journal of Graph Algorithms and Applications, Tome 15 (2011) no. 4, pp. 485-501. doi : 10.7155/jgaa.00235. http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00235/

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