Lights Out for graphs related to one another by constructions

Abstract

The Lights Out problem on graphs, in which each vertex of the graph is in one of two states (“on” or “off”), has been investigated from a variety of perspectives over the last several decades, including parity domination, cellular automata, and harmonic functions on graphs. We consider a variant of the Lights Out problem in which the possible states for each vertex are indexed by the integers modulo $k$. We examine the space of “null patterns” (i.e., harmonic functions) on graphs, and use this as a way to prove theorems about Lights Out on graphs that are related to one another by two main constructions.