Given a graph G with positive integer weights on the vertices, and a token placed on some current vertex u, two players alternately remove a positive integer weight from u and then move the token to a new current vertex adjacent to u. When the weight of a vertex is set to 0, it is removed and its neighborhood becomes a clique. The player making the last move wins. This adaptation of Nim on graphs is called Vertexnim, and slightly differs from the game Vertex NimG introduced by Stockman in 2004. Vertexnim can be played on both directed or undirected graphs. In this paper, we study the complexity of deciding whether a given game position of Vertexnim is winning for the first or second player. In particular, we show that for undirected graphs, this problem can be solved in quadratic time. Our algorithm is also available for the game Vertex NimG, thus improving Stockman's exptime algorithm. In the directed case, we are able to compute the winning strategy in polynomial time for several instances, including circuits or digraphs with self loops.
Univ Lyon, Université Lyon 1, LIRIS UMR CNRS 5205（里昂第一大学）；Instituto de Matemáticas, UNAM Juriquilla（墨西哥国立自治大学数学研究所）；National University of Singapore（新加坡国立大学）
A probabilistic version of the game of Zombies and Survivors on graphs
我们考虑一个新的概率图搜索游戏的图形，灵感来自熟悉的游戏警察和强盗。在Zombies and Survivors中，一组僵尸试图在给定的图形上吃掉一个孤独的幸存者。僵尸随机选择他们的初始位置，并在游戏过程中，直接向幸存者移动。在每一轮，他们移动到邻近的顶点，最小化到幸存者的距离;如果有一个以上的这样的顶点，那么他们随机选择一个。幸存者试图通过移动到邻近的顶点或留在他当前的顶点来逃离僵尸。如果最终有一个僵尸落在它们的顶点上吃掉了幸存者，僵尸就赢了;否则，幸存者就赢了。图的僵尸数是游戏所需的僵尸的最小数量，使得它们获胜的概率严格大于1/2。我们提出的僵尸数的几个图形的家庭，如周期，超立方体，投影平面的关联图，笛卡尔和环形网格的渐近结果。
