Extremal Graphs for the Lights Out Problem

Lights Out is a game played on a graph $G$ where every vertex has a light bulb that is either on or off, and pressing a vertex $v$ toggles the state of every vertex in the closed neighborhood of $v$. The goal is to find a subset of vertices $S$ such that pressing every vertex in $S$ results in all light bulbs being turned off. We study the extremal graphs for which pressing every vertex is the unique solution to the lights out problem given an initial configuration of all lights on. We show that a graph is extremal if and only if it is even and has an odd number of matchings. Furthermore, there is a bijection between the set of labeled $n$-vertex extremal graphs and the set of symmetric invertible matrices of size $n-2$ over $\mathbb{F}_2$. We prove that any even graph with no cycle of length $0\pmod 3$ must be extremal. We also demonstrate operations that build larger extremal graphs from smaller ones. Along the way, we prove using the polynomial method that in any even graph, the number of matchings of a fixed size covering an odd subset of vertices is even.

💡 Research Summary

The paper investigates a special class of graphs for the Lights Out puzzle, namely those for which the unique solution when all lights start on is to press every vertex. Such graphs are called extremal. The authors provide a complete combinatorial and algebraic characterization, enumerate them exactly, and develop constructive operations that generate larger extremal graphs from smaller ones.

Main Results

1. Characterization via parity and matchings (Theorem 2.1).
   Let (M_G = A_G + I_n) be the adjacency matrix of a graph (G) over the field (\mathbb{F}_2). The system (M_G x = \mathbf{1}) encodes the Lights Out condition. The authors show that (\det(M_G) \equiv m(G) \pmod 2), where (m(G)) is the total number of matchings (including the empty matching). Consequently, a graph is extremal if and only if (i) it is even (every closed neighbourhood has even size, which guarantees that (\mathbf{1}) lies in the column space of (M_G)) and (ii) (m(G)) is odd (which forces (\det(M_G)=1) and thus the solution is unique). This bridges the puzzle with classical graph parity and matching theory.

2. Exact counting via a matrix bijection (Theorem 3.2).
   Extremal graphs on a labeled vertex set (