The Infectious Vaccination Problem: a variant of Firefighting with Spreading Defence

2026-10-04

AI Summary

This study addresses a spreading-defense variant of the classical Firefighter problem, in which defensive measures propagate across the graph rather than remaining confined to single vertices. Motivated by the aerial deployment of rabies vaccines for bat populations, the work investigates the computational challenge of determining optimal defense strategies when protection can diffuse. To overcome the limitations of static defense models, the authors introduce a “contagious” defense mechanism and employ an integrated methodology combining graph-theoretic algorithms, computational complexity theory, and mathematical analysis on infinite-dimensional Cartesian and strong grid graphs. The primary contributions include establishing an algorithmic framework and delineating computational complexity bounds for the spreading defense problem, proving structural properties on specific classes of infinite grid graphs, and extending the theoretical boundaries for modeling propagation processes on graphs.

Abstract

The Firefighter Problem models a spreading process (originally a fire, alternatively an infection or rumour, for example) on a graph. A defender saves a single vertex per turn; after each defence, the fire spreads to the unburned and undefended neighbours of all burning vertices. Deciding whether a strategy exists for the defender to protect some targeted number of vertices is computationally hard in graphs in general, but tractable in some restricted cases. Inspired by research into spreadable rabies vaccines for bats, we study a variant of the Firefighter problem in which defence also spreads. Some approximation results are already known for this problem; we provide algorithmic and hardness results, as well as containment results for the infinite $n$-dimensional Cartesian and strong grid graphs.

Problem

• Firefighter Problem
• Infectious Vaccination
• Spreading Defence
• Graph containment
• Computational hardness

Innovation

• Firefighter Problem
• Infectious Vaccination
• Spreading Defence
• Grid Graphs
• Containment