Liminal Burning in Graph Diffusion Games

• Liminal burning is a graph diffusion model where a Saboteur reveals k vertices per round, constraining the Arsonist’s source choices and defining the k-liminal burning number bₖ(G).
• It interpolates between ordinary graph burning (unrestricted ignition) and cooling (single vertex revelation), offering insights into adversarial diffusion dynamics.
• Exact formulas and bounds on paths, grids, and hypercubes illustrate its unique geometric constraints and computational challenges in optimal source placement.

Liminal burning is a graph-theoretic diffusion game that generalizes both ordinary graph burning and graph cooling. In $k$-liminal burning, a Saboteur reveals $k$-sets of vertices in each round, and the Arsonist may ignite new sources only inside the revealed set, while ordinary fire propagation continues as in standard graph burning. The resulting parameter, the $k$-liminal burning number $b_k(G)$, measures how quickly a network can be burned under adversarially constrained source placement. Recent work treats liminal burning as a two-player perfect knowledge game and places it explicitly “between burning and cooling” (Bonato et al., 15 May 2025, Ambrose et al., 24 Sep 2025).

1. Formal graph-theoretic model

Liminal burning is built on the standard graph burning process. For a finite connected graph $G$, ordinary burning proceeds in discrete rounds: one new source may be ignited each round, and fire from previously burned vertices spreads to all neighbors. A sequence $(x_1,\dots,x_k)$ burns $G$ in $k$ rounds exactly when

$$V(G)=N_{k-1}[x_1]\cup N_{k-2}[x_2]\cup \cdots \cup N_0[x_k],$$

and the minimum such $k$ is the burning number of $k$0 (Farokh et al., 2020).

In liminal burning, vertices carry two binary states: burned or unburned, and revealed or unrevealed. For a fixed liminal number $k$1, the game on a finite connected graph $k$2 proceeds as follows. First, the Saboteur reveals $k$3 vertices, which remain revealed for the entire game. The Arsonist then burns one revealed vertex. On each later turn, fire spreads from already burned vertices to their neighbors, the Saboteur reveals $k$4 additional vertices, and the Arsonist chooses another revealed vertex to burn as a new source. The game ends when all vertices are burned. The Saboteur seeks to maximize the number of turns, while the Arsonist seeks to minimize it; under optimal play, this value is the $k$5-liminal burning number $k$6 (Ambrose et al., 24 Sep 2025).

This formulation makes liminal burning a constrained source-placement problem. Ordinary burning asks where to place sources in space and time so that expanding metric balls cover the graph efficiently. Liminal burning adds an adversarial revelation interface: the Arsonist cannot choose arbitrary ignition sites, but only those made available by the Saboteur. This suggests a controlled-spread interpretation in which local contagion propagates autonomously, while global reseeding is filtered by an opposing agent.

2. Interpolation between burning and cooling

The central structural feature of liminal burning is that it interpolates between two previously studied processes. For $k$7, liminal burning is identical to ordinary graph burning, because every vertex is effectively available as a possible source. For $k$8, liminal burning is identical to cooling, because the Saboteur determines the only available source at each round (Bonato et al., 15 May 2025, Ambrose et al., 24 Sep 2025).

The basic parameters can be organized as follows.

Symbol	Meaning	Extremal interpretation
$k$9	Ordinary burning number	Unrestricted source choice
$k$0	$k$1-liminal burning number	Saboteur reveals $k$2 vertices per round
$k$3	Cooling number	$k$4
$k$5	Smallest $k$6 with $k$7	Robustness threshold

The threshold parameter

$k$8

measures how much revelation is needed before the Saboteur’s restrictions no longer increase the burning time. In this sense, $k$9 is a robustness threshold for ordinary burning under adversarial revelation constraints (Ambrose et al., 24 Sep 2025).

The path of values between cooling and burning is encoded by the chain

$b_k(G)$0

as stated in the liminal-burning literature (Ambrose et al., 24 Sep 2025). This sequence formalizes the interpolation, even though the indexing convention is unusual from the perspective of source availability.

3. Path graphs and the threshold parameter $b_k(G)$1

Paths are the first graph family for which liminal burning has exact nontrivial formulas. For ordinary burning,

$b_k(G)$2

so path graphs provide the canonical benchmark for comparing adversarial revelation against unrestricted burning (Guillaume et al., 2 Jun 2026).

A principal exact result is

$b_k(G)$3

The proof uses an explicit Saboteur strategy: reveal the leftmost available vertices at each turn, forcing the fire front to advance at a controlled rate. The resulting dynamics burn one vertex on the first turn and then three vertices per later turn, up to the final partial round (Ambrose et al., 24 Sep 2025).

The same paper introduces the main threshold theorem for square-length paths. For $b_k(G)$4, if $b_k(G)$5 is the smallest $b_k(G)$6 such that

$b_k(G)$7

and if

$b_k(G)$8

with

$b_k(G)$9

then

$G$0

The underlying idea is that optimal ordinary burning of $G$1 in $G$2 turns corresponds to tiling the path by odd intervals of lengths $G$3, and $G$4 depends on how many placements of the largest interval remain feasible under adversarial revelation (Ambrose et al., 24 Sep 2025).

These results establish two enduring themes. First, liminal burning on paths is already substantially different from ordinary burning: the exact formula for $G$5 is linear in $G$6, whereas $G$7 is $G$8. Second, the threshold $G$9 is inherently combinatorial, because it depends on the existence of compatible interval tilings rather than only on path length. Exact formulas for $(x_1,\dots,x_k)$0 for general $(x_1,\dots,x_k)$1 remain an open direction explicitly identified in the literature (Ambrose et al., 24 Sep 2025).

4. Hypercubes, grids, and other structured families

The dedicated 2025 study of liminal burning reports a broader program beyond paths. Using a variant of Sperner sets, it studies $(x_1,\dots,x_k)$2-liminal burning numbers of hypercubes and gives bounds and exact values for various values of $(x_1,\dots,x_k)$3. In particular, it determines the exact cooling number of the $(x_1,\dots,x_k)$4-dimensional hypercube to be $(x_1,\dots,x_k)$5. It also analyzes liminal burning for Cartesian grids and products, paths, and graphs whose vertex sets can be decomposed into many components of small diameter (Bonato et al., 15 May 2025).

The visible grid analysis develops explicit delaying strategies for the Saboteur. On a grid $(x_1,\dots,x_k)$6, the brute-force Saboteur strategy reveals the $(x_1,\dots,x_k)$7 vertices closest to $(x_1,\dots,x_k)$8 during each round, with ties broken by choosing $(x_1,\dots,x_k)$9 before $G$0 for $G$1. One of the resulting lemmas states that, if

$G$2

then for at most

$G$3

rounds, no vertex at distance $G$4 or more from $G$5 is revealed or burned (Bonato et al., 15 May 2025). The subsequent grid arguments use differential-inequality estimates for the volume of the burned set and a symmetry transfer from $G$6 to $G$7, indicating a multi-phase theory of delayed spread on Cartesian grids.

This suggests that liminal burning has a strong geometric component on structured product graphs. On paths, the obstruction comes from interval tilings. On grids and hypercubes, the obstruction is spatial: the Saboteur can localize revealed vertices near a corner or along a geometric frontier, forcing the Arsonist to spend many rounds without reaching distant regions.

5. Complexity and algorithmic status

Liminal burning is computationally hard at the level of game value. The 2025 paper shows that liminal burning a graph is PSPACE-complete for $G$8, using a reduction from $G$9-QBF. It also proves, through a reduction from burning, that even in some cases when liminal burning is likely not PSPACE-complete, it is co-NP-hard (Bonato et al., 15 May 2025).

These results sit on top of an already difficult one-player theory. Ordinary graph burning is NP-complete, and the decision problem remains NP-complete even on restricted classes such as trees of maximum degree $k$0, spider graphs, and path-forests (Farokh et al., 2020). Later work also shows NP-completeness for connected cubic graphs and connected proper interval graphs (Antony et al., 2024). Since liminal burning introduces adversarial revelation on top of ordinary burning, the jump from NP-completeness to PSPACE-completeness reflects the passage from static optimization to optimal play in a finite perfect-information game.

The algorithmic landscape is therefore split. Exact formulas are known on selected graph families, especially paths, and the dedicated 2025 study reports additional exact or bounded results for hypercubes and structured products (Bonato et al., 15 May 2025). But the general decision problem is intractable in the strong game-theoretic sense. A plausible implication is that liminal burning should be viewed less as a single optimization parameter and more as a family of adversarial diffusion problems whose tractable cases depend heavily on graph structure.

6. Related adversarial and threshold-burning models

Liminal burning belongs to a wider cluster of graph-burning variants in which thresholds, adversaries, or constrained propagation alter the classical model. One nearby example is adversarial graph burning on growing graphs. There, Builder adds $k$1 new vertices at turn $k$2, Arsonist burns one new source per round, and the objective is asymptotic burned density rather than finite completion time. For polynomial growth $k$3, the paper proves a threshold: $k$4 Thus linear growth is the critical order separating asymptotic engulfment from persistent non-burning mass (Gunderson et al., 2022).

Another related but distinct threshold model is lazy burning on hypergraphs. There, one chooses an initial burned set only once, and a vertex burns later exactly when it is the unique unburned vertex in some hyperedge. The lazy burning number satisfies the exact formula

$k$5

where $k$6 is the maximum cardinality of a $k$7-matching in the incidence graph of $k$8 (Bonato et al., 2024). This process is not liminal burning in the game-theoretic sense, but it is closely related in spirit because propagation occurs at a sharp all-but-one threshold.

These neighboring models clarify the conceptual position of liminal burning. It is not merely ordinary burning with randomness, and it is not the same as growth-versus-burning density games or hypergraph threshold ignition. Its distinctive feature is adversarial control of source availability under otherwise standard burning dynamics. Within graph-burning theory, that places liminal burning at the interface between diffusion, game complexity, and threshold-controlled spread.