# Online Firefighting on Grids

## TL;DR

This paper extends the online Firefighter Problem to infinite square grids, providing conditions and strategies for fire containment when firefighter resources are revealed in real-time.

## Contribution

It introduces a set of sufficient conditions and strategies for online fire containment specifically on infinite square grids, expanding the problem's scope beyond trees.

## Key findings

- Provided conditions for fire containment on infinite grids
- Developed strategies for online firefighting in grid graphs
- Extended the online Firefighter Problem to new graph classes

## Abstract

The Firefighter Problem (FP) is a graph problem originally introduced in 1995 to model the spread of a fire in a graph, which has attracted considerable attention in the literature. The goal is to devise a strategy to employ a given sequence of firefighters on strategic points in the graph in order to contain efficiently the fire (which spreads from each unprotected vertex to all of it neighbours on successive time steps). Recently, an online version of FP---where the number of firefighters available at each turn are revealed in real-time--- has been introduced in the literature and studied on trees. In this paper, we consider the online containment of fire on square grids. In particular, we provide a set of sufficient conditions that allow to solve the online version of the firefighting problem on infinite square grids, illustrating the corresponding fire containment strategies.