The Containment Game in the plane: between the Firefighter Problem and Conway's Angel Problem

TL;DR

The work introduces the Containment Game, a two-player generalization bridging Conway's Angel problem and the Firefighter problem, and analyzes how sublinear spreading $g(t)$ can match the Firefighter containment difficulty on planar graphs. The authors construct explicit Spreader strategies on the Eighth Plane, the Directed Half-Plane, and then the full plane, employing a novel potential- and debt-based bookkeeping to prove sublinear upper bounds: $g(t)=O(t^{6/7})$, while also establishing a lower bound $g(t)=\Omega(t^{1/2})$ below which defeating Spreader remains easier than the Firefighter solitaire. The proofs hinge on decomposing the strategy into segments, controlling path counts with $t$-paths, and ensuring sparsity via careful doubling times and look-ahead analyses; these ideas extend to directed variants and enable a unified framework across multiple graph settings. Collectively, the results demonstrate that sublinear spreading can, in fact, emulate unrestricted fire growth in these planar graphs, advancing understanding of containment thresholds and informing related pursuit-evasion models on infinite graphs.

Abstract

The containment game is a full information game for two players, initialised with a set of occupied vertices in an infinite connected graph $G$. On the $t$-th turn, the first player, called Spreader, extends the occupied set to $g(t)$ adjacent vertices, and then the second player, called Container, removes $q$ unoccupied vertices from the graph. If the spreading process continues perpetually -- Spreader wins, and otherwise -- Container wins. For $g=\infty$ this game reduces to a solitaire game for Container, known as the Firefighter Problem. On $\mathbb{Z}^2$, for $q=1/k$ and $g\equiv 1$ it is equivalent to Conway's Angel Problem. We introduce the game, and writing $q(G,g)$ for the set of $q$ values for which Container wins against a given $g(t)$, we study the minimal asymptotics of $g(t)$ such that $q(G,g)=q(G,\infty)$, i.e. for which defeating Spreader is as hard as winning the Firefighter Problem solitaire. We show, by providing explicit winning strategies, a sub-linear upper bound $g(t)=O(t^{6/7})$ and a lower bound of $g(t)=Ω(t^{1/2})$.

Figures (11)

• Figure 1: Two possible cases for the location of $(a,t)$ within $S$, and the corresponding paths. In each sub-figure, the central path $P$ is indicated by an unbroken line, while the other paths in are indicated by dashed lines.
• Figure 2: Illustration of different phenomena of the main Spreader strategy. The illustrated phenomena are presented in different phases, marked by horizontal dashed lines, while the partition into segments is marked by vertical lines. In Phase I, all segments, being close to the boundary, are spreading. In Phase II, the inner segments become simulative, and their occupied vertices consolidate to single occupied vertices. In Phase III, the occupied vertices move in response to deleted vertices, those possessing an unblocked path move along it, while those who do not cease playing. All deleted vertices of this phase were removed at least $h$ vertices away from the front, hence simulative segments in their vicinity remain simulative. In Phase IV, two disruptions occur, one after another. This causes the segments in their vicinity to become spreading for a while, until they consolidate once again at the end of the step. In Phase V, the right boundary is blocked by non-disruptive deleted vertices. This causes the segments at the boundary to become spreading, such that at the end of the step there is still a bulk of occupied vertices at the boundary. Phase VI opens with a doubling. In the boundary this causes some segments to become spreading, while the inner simulative segments have half of their vertices discontinued.
• Figure 3: Depicted in each sub-figure, are a segment and the union of its look-ahead regions in in two consecutive time-steps $t-1$ and $t$. Look-ahead distances are marked by vertical lines and $S\times L_{t-1}(S)$ by a filling pattern. In \ref{['fig:segment step, simulated']} the segment is simulative, in \ref{['fig:segment step, transitional']} it is spreading with positive look-ahead, while in \ref{['fig:segment step, simple']} it is simple.
• Figure 4: \ref{['fig:plane, initial game']} illustrates the initial game in the plane. The four fronts are outlined by rectangles, and the four disjoint infinite trapezoids are outlined by dashed lines. The infinite trapezoid of front $0$ is highlighted by a filling pattern. In \ref{['fig:plane, spilling']} front $0$ is re-ignited by front $1$. The initial occupied set of the re-initialised game at front $0$ is depicted in gray, and the infinite trapezoid is highlighted by a filling pattern.
• Figure 5: Step 1

Theorems & Definitions (63)

• Theorem 1
• Theorem 2
• Remark 1
• Remark 2
• Claim 2.1
• proof
• Claim 2.2
• proof
• Proposition 2.7
• proof