An Extremal Problem for the Neighborhood Lights Out Game

Neighborhood Lights Out is a game played on graphs. Begin with a graph and a vertex labeling of the graph from the set $\{0,1,2,\dots, \ell-1\}$ for $\ell \in \mathbb{N}$. The game is played by toggling vertices: when a vertex is toggled, that vertex and each of its neighbors has its label increased by $1$ (modulo $\ell$). The game is won when every vertex has label 0. For any $n\in\mathbb{N}$ it is clear that one cannot win the game on $K_n$ unless the initial labeling assigns all vertices the same label. Given that the $K_n$ has the maximum number of edges of any simple graph on $n$ vertices it is natural to ask how many edges can be in a graph so that the Neighborhood Lights Out game is winnable regardless of the initial labeling. We find all such extremal graphs on $n$ vertices that have $\binom{n}{2} - c$ edges for $c\leq \lceil\frac{n}{2}\rceil +3$ and all those that have minimum degree $n-3$. The proofs of our results require us to introduce a new version of the Lights Out game that can be played given any square matrix.