Lights Out: The Effect of Graph Operations on Null Space Dimension
Dr. Darin Stephenson
Based on a handheld puzzle created by Tiger Electronics, Lights Out is a problem in graph theory and linear algebra that has been investigated for over 30 years. A graph begins with vertices that are either on or off, and the edges determine which other vertices are affected when just one vertex is turned on or off. The objective of Lights Out is to turn off all of the lights, in which case a graph has been won. Our approach to the Lights Out problem has not only been on the investigation of winnable states of graphs, but has also led to the investigation of the null space of the neighborhood matrices of those graphs, and how the null space changes when a graph is altered. We develop results that determine how the Lights Out puzzle is affected when two or more graphs are connected together in various ways, or when certain subgraphs are removed.
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