For a graph $G$ and an integer-valued function $\tau$ on its vertex set, a dynamic monopoly is a set of vertices of $G$ such that iteratively adding to it vertices $u$ of $G$ that have at least $\tau(u)$ neighbors in it eventually yields the vertex set of $G$. We study the problem of maximizing the minimum order of a dynamic monopoly by increasing the threshold values of individual vertices subject to vertex-dependent lower and upper bounds, and fixing the total increase. We solve this problem efficiently for trees, which extends a result of Khoshkhah and Zaker (On the largest dynamic monopolies of graphs with a given average threshold, Canadian Mathematical Bulletin 58 (2015) 306-316).
翻译:对于一个G$图和其顶端的整数值函数$tau美元,动态垄断是一套G$的顶端,它反复增加顶端为$1美元,其中邻居至少拥有$tau(u)美元,最终产生顶端为$1美元。我们研究如何通过提高受顶端依赖的下界和上界个体顶端的顶端的顶端值和确定总增加量来最大限度地增加动态垄断的最低顺序的问题。我们有效地解决了树木问题,这是Khoshkhah和Zaker(关于具有给定平均阈值的最大动态图的垄断,加拿大数学公报58/2015 306-316)的结果。