标题： 图上的一个新的支配集博弈 显示英文标题

标题： A New Dominating Set Game on Graphs

摘要： 我们引入了一个新的图上的双人游戏，玩家轮流选择顶点，直到所选顶点的集合形成一个支配集。最后一个选择顶点的玩家获胜。这个游戏符合其他几种已知图上游戏的模式。我们对具有先手必胜策略的路径和环进行了表征。我们还创建了各种组合图的工具（通过图幂、笛卡尔乘积、图联接和其他方法），用于构建各种由第二位玩家获胜的图，包括立方体、奇数个顶点的多维网格、大多数多维环面网格、各种树如特殊毛虫树、彼得森图等。最后，我们将该游戏扩展到群，并证明在具有规范生成集的偶阶交换群上，第二位玩家获胜，以及其他情况。 显示英文摘要

摘要： We introduce a new two-player game on graphs, in which players alternate choosing vertices until the set of chosen vertices forms a dominating set. The last player to choose a vertex is the winner. The game fits into the scheme of several other known games on graphs. We characterize the paths and cycles for which the first player has the winning strategy. We also create tools for combining graphs in various ways (via graph powers, Cartesian products, graph joins, and other methods) for building a variety of graphs whose games are won by the second player, including cubes, multidimensional grids with an odd number of vertices, most multidimensional toroidal grids, various trees such as specialized caterpillars, the Petersen graph, and others. Finally, we extend the game to groups and show that the second player wins the game on abelian groups of even order with canonical generating set, among others.