Vaccinate your trees!

TL;DR

This paper investigates vaccination strategies in graphs to maximize the size of dynamic monopolies, providing efficient solutions for trees by adjusting thresholds or removing vertices within a given budget.

Contribution

It introduces methods to optimize dynamic monopoly sizes in trees through threshold adjustments and vertex removals, addressing vaccination problems in graph theory.

Findings

Efficient algorithms for trees to maximize dynamic monopoly size.

Strategies for threshold increases and vertex removals within a budget.

Theoretical insights into vaccination problems in graphs.

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 two vaccination problems, where the goal is to maximize the minimum order of such a dynamic monopoly either by increasing the threshold value of $b$ vertices beyond their degree, or by removing $b$ vertices from $G$, where $b$ is a given non-negative integer corresponding to a budget. We show how to solve these problems efficiently for trees.