On Generalizing the “Lights Out” Game and a Generalization of Parity Domination

Abstract

The Lights Out game on a graph \(G\) is played as follows. Begin with a (not necessarily proper) coloring of \(V(G)\) with elements of \(\mathbb{Z}_2\). When a vertex is toggled, that vertex and all adjacent vertices change their colors from \(0\) to \(1\) or vice-versa. The game is won when all vertices have color \(0\). The winnability of this game is related to the existence of a parity dominating set.
We generalize this game to \(\mathbb{Z}_k\), \(k \geq 2\), and use this to define a generalization of parity dominating sets. We determine all paths, cycles, and complete bipartite graphs in which the game over \(\mathbb{Z}_k\) can be won regardless of the initial coloring, and we determine a constructive method for creating all caterpillar graphs in which the Lights Out game cannot always be won.