Between burning and cooling: liminal burning on graphs

TL;DR

This work introduces liminal burning, a parameterized two-player game on graphs that interpolates between burning and cooling via $b_k(G)$. It develops exact and bounds results for hypercubes, analyzes Cartesian grids and general products, and links $b_k(G)$ to rainbow Sperner families and hypergraph matchings. A central contribution is showing PSPACE-completeness of computing $b_k(G)$ for all $k\ge 2$, complemented by structural upper bounds from $(k,d)$-special graphs and hypergraph edge covers. The results illuminate how graph structure and product operations shape the liminal burning dynamics, and they lay out several open problems and future directions in complexity, random graphs, and broader graph families.

Abstract

Liminal burning generalizes both the burning and cooling processes in graphs. In $k$-liminal burning, a Saboteur reveals $k$-sets of vertices in each round, with the goal of extending the length of the game, and the Arsonist must choose sources only within these sets, with the goal of ending the game as soon as possible. The result is a two-player game with the corresponding optimization parameter called the $k$-liminal burning number. For $k = |V(G)|$, liminal burning is identical to burning, and for $k = 1$, liminal burning is identical to cooling. Using a variant of Sperner sets, $k$-liminal burning numbers of hypercubes are studied along with bounds and exact values for various values of $k$. In particular, we determine the exact cooling number of the $n$-dimensional hypercube to be $n.$ We analyze liminal burning for several graph families, such as Cartesian grids and products, paths, and graphs whose vertex sets can be decomposed into many components of small diameter. We consider the complexity of liminal burning and show that liminal burning a graph is PSPACE-complete for $k\geq 2,$ using a reduction from $3$-QBF. We also prove, through a reduction from burning, that even in some cases when liminal burning is likely not PSPACE-complete, it is co-NP-hard. We finish with several open problems.

Figures (4)

• Figure 1: An example of a $2$-liminal burning game played on the $5 \times 5$ Cartesian grid. Sets, in order, are revealed by the Saboteur and marked in blue outlines, while vertices burned by the Arsonist are marked in red.
• Figure 2: The upper and lower bounds of $b_k(G_{100})$, along with a heuristic computational solution. The vertical dashed lines represent the transition from one range of $k$ to the next.
• Figure 3: The graph $G_\phi$ when $\phi(x_1,x_2,x_3)=(x_1 \lor x_2 \lor \lnot x_3) \land (\lnot x_1 \lor \lnot x_2 \lor x_3)$.
• Figure 4: The graph $H_4$.

Theorems & Definitions (58)

• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof