Abstract A subset S of vertices in a graph G = ( V , E ) is called a global offensive alliance if for every vertex v not in S , at least half of the vertices in the closed neighborhood of v are in S . A global offensive alliance D is called a global strong offensive alliance if for every vertex v not in S , more than half of the vertices in the closed neighborhood of v are in S . The global offensive alliance number (global strong offensive alliance number, respectively) of G is the minimum cardinality of a global offensive alliance (global strong offensive alliance, respectively) in G . In this paper, we present new (probabilistic) upper bounds for the global offensive alliance number as well as the global strong offensive alliance number of a graph, improving previous bounds given in Harutyunyan (2014).

中文翻译：

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图表中全球进攻联盟的上限

摘要 如果对于不在 S 中的每个顶点 v ，在 v 的封闭邻域中至少有一半的顶点在 S 中，则图 G = ( V , E ) 中顶点的子集 S 被称为全局进攻联盟。如果对于每个不在 S 中的顶点 v，v 的封闭邻域中的一半以上的顶点都在 S 中，则全局进攻联盟 D 被称为全球强进攻联盟。G 的全球进攻联盟数（分别为全球强进攻联盟数）是 G 中全球进攻联盟（分别为全球强进攻联盟）的最小基数。在本文中，我们提出了图的全球进攻联盟数量以及全球强进攻联盟数量的新（概率）上限，改进了 Harutyunyan (2014) 中给出的先前边界。