Partial immunization of trees

TL;DR

This paper investigates how to optimally increase vertex thresholds in a graph to maximize the smallest dynamic monopoly size, providing an efficient solution specifically for trees, extending prior results.

Contribution

It introduces an efficient method for maximizing the minimum dynamic monopoly in trees by adjusting thresholds within given bounds, extending previous work.

Findings

Efficient solution for trees to maximize minimum dynamic monopoly size.

Extension of prior results to a broader class of graphs.

Provides a framework for threshold adjustment under constraints.

Abstract

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).