For a graph $G$ and a non-negative 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 $\tau \left(u\right)$ 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 (2015).

中文翻译：

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树木的部分免疫

对于图 $G$ 和一个非负整数值函数 $\tau$ 在其顶点集上，动态垄断是一组顶点 $G$ 这样反复添加顶点 $ü$ 的 $G$ 至少有 $\tau （ü）$ 最终它的邻居产生了一个顶点集 $G$。我们研究了通过增加受顶点依赖的上下限限制的各个顶点的阈值并固定总增量来最大化动态垄断的最小顺序的问题。我们有效地为树木解决了这个问题，这是Khoshkhah and Zaker（2015）的结果。