Firefighting with a Distance-Based Restriction

TL;DR

This paper introduces the distance-restricted firefighting model, constraining each firefighter to move at most distance $d$ per turn and formalizing containment via a bipartite matching framework. It compares distance-restricted containment with the classic game, showing that monotonicity with respect to subgraphs can fail and that the distance-restricted firefighter number $f_d(G)$ can behave very differently from the unrestricted case. The authors provide detailed results for infinite grids: on the square grid, $d=1$ requires four firefighters and $d\ge3$ requires two, while the $d=2$ case remains unsettled with $2\le f_2\le 3$; on the strong grid, $f_1=8$ and $f_2=4$; on the hexagonal grid, $d=1$ requires three and $d\ge2$ requires two, with a conjecture about $d=2$. They also pose open problems and conjectures, outlining approaches toward proving Messinger-type results and highlighting structural questions about containment under distance restrictions.

Abstract

In the classic version of the game of firefighter, on the first turn a fire breaks out on a vertex in a graph $G$ and then $k$ firefighters protect $k$ vertices. On each subsequent turn, the fire spreads to the collective unburnt neighbourhood of all the burning vertices and the firefighters again protect $k$ vertices. Once a vertex has been burnt or protected it remains that way for the rest of the game. A common objective with respect to some infinite graph $G$ is to determine how many firefighters are necessary to stop the fire from spreading after a finite number of turns, commonly referred to as containing the fire. We introduce the concept of distance-restricted firefighting where the firefighters' movement is restricted so they can only move up to some fixed distance $d$ per turn rather than being able to move without restriction. We establish some general properties of this new game in contrast to properties of the original game, and we investigate specific cases of the distance-restricted game on the infinite square, strong, and hexagonal grids. We conjecture that two firefighters are insufficient on the square grid when $d = 2$, and we pose some questions about how many firefighters are required in general when $d = 1$.

Figures (14)

• Figure 2.1: $7 \times 7$ Portion of the infinite hexagonal grid as a subgraph of the infinite square grid.
• Figure 2.2: $7 \times 7$ Portion of sub($G_{\hexagon}$) as a subgraph of the infinite hexagonal grid.
• Figure 2.3: A strategy to contain the fire in four turns on sub($G_{\hexagon}$) when $d=11$. The case where the fire begins on a degree two vertex is trivial and thus omitted.
• Figure 2.4: Left: The ball of radius $3$ around the initial burning vertex of $P^{\mathbb{N}}\:\square\:C^5$ modified as described in the proof of Theorem \ref{['thm:arb_diff_2']}, Right: An infinite subgraph of the graph on the left with distance-restricted firefighter number $5$.
• Figure 3.1: A strategy for containment on the infinite square grid with two firefighters when $d=3$. Recall that burnt vertices are round and orange whereas defended vertices are square and black, except in round zero where they are diamond and star shaped respectively.

Theorems & Definitions (27)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Corollary 2.3
• Corollary 2.4
• Lemma 3.1
• proof
• Theorem 3.2
• proof