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

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 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 Z(k), k >= 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 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.