A characterization of always solvable trees in the Lights Out game using the activation types of vertices

Abstract

Lights out is a game that can be played on any simple graph $G$. A configuration assigns one of the two states \emph{on} or \emph{off} to each vertex. For a given configuration, the aim of the game is to turn all vertices \emph{off} by applying a push pattern on vertices, where each push switches the state of the vertex and its neighbors. If every configuration of vertices is solvable, then we say that the graph is always solvable. We introduce a concept which we call the activation types of vertices and we prove several characterization results of trees by using this concept. We showed that all always solvable trees different than the star tree can be seen as the join graph of its two always solvable subtrees. We call the dimension of the space of null-patterns, which leave configurations unchanged, the nullity of the graph $G$. We show that the nullity of a tree can be characterized by the cardinality of its minimal partition into always solvable subtrees. We also showed that nullity of a tree is less than the number of its even degree vertices.