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Lights out is a game which can be played on any graph $G$. Initially we havea configuration which assigns one of the two states on or off to each vertex.The aim of the game is to turn all vertices to off state for an initialconfiguration by activating some vertices where each activation switches thestate of the vertex and all of its neighbors. If the aim of the game can beaccomplished for all initial configurations then $G$ is called always solvable.We call the dimension of the kernel of the closed neighborhood matrix of thegraph over the field $\mathbb{Z}_2$, nullity of $G$. It turns out that $G$ isalways solvable if and only if its nullity is zero. We investigate the problemof how nullity changes when an edge is added to or removed from a graph. As aresult we characterize always solvable graphs. We also showed that for everygraph with positive nullity there exists an edge whose removal decreases thenullity. Conversely, we showed that for every always solvable graph which isnot an even graph with odd order, there exists an edge whose addition increasesthe nullity.

Effects of edge addition or removal on the nullity of a graph in the Lights Out game