Steiner三元系上nofil的组合对策
The combinatorial game nofil played on Steiner Triple Systems
摘要：
我们介绍了一个公平的组合对策的Steiner三元系称为Nofil。玩家交替移动，选择三元系的点数。如果一个玩家被迫在他们的回合中填满一个方块，他们就输了。我们探讨发挥诺菲尔所有施泰纳三元系的顺序为15和抽样订单19，21和25。我们通过计算每个博弈及其子博弈的nim值来确定最优策略。游戏Nofil可以被认为是在一个相应的超图上玩。随着游戏的进行，超图会缩小，最终将等同于在同构图上玩游戏Node Kayles。Node Kayles是很好的研究和理解。出于这一动机，我们研究节点凯尔斯的位置可以达到，即嵌入到一个施泰纳三元系。我们证明了存在这种图嵌入的必要条件和充分条件，并得出结论，确定游戏的结果Nofil Steiner三元系的复杂性是PSPACE-完全的。
英文摘要：
We introduce an impartial combinatorial game on Steiner triple systems called Nofil. Players move alternately, choosing points of the triple system. If a player is forced to fill a block on their turn, they lose. We explore the play of Nofil on all Steiner triple systems up to order 15 and a sampling for orders 19, 21, and 25. We determine the optimal strategies by computing the nim-values for each game and its subgames. The game Nofil can be thought of in terms of play on a corresponding hypergraph. As game play progresses, the hypergraph shrinks and will eventually be equivalent to playing the game Node Kayles on an isomorphic graph. Node Kayles is well studied and understood. Motivated by this, we study which Node Kayles positions can be reached, i.e. embedded into a Steiner triple system. We prove necessary conditions and sufficient conditions for the existence of such graph embeddings and conclude that the complexity of determining the outcome of the game Nofil on Steiner triple systems is PSPACE-complete.
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