Lights Out on finite graphs

Edwards, Stephanie; Elandt, Victoria; James, Nicholas; Johnson, Kathryn; Mitchell, Zachary; Stephenson, Darin R.

doi:10.2140/involve.2010.3.17

Cited by 8 publications

(4 citation statements)

References 4 publications

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“…To determine winnability, we depend heavily on linear algebra methods similar to those in [2,5,9], and [10]. We discuss these methods in Section 2.…”

Section: An Extremal Problem For the Neighborhood Lights Out Game 999mentioning

confidence: 99%

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An extremal problem for the neighborhood Lights Out game

Keough

Parker

2024

Discuss. Math. Graph Theory

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Neighborhood Lights Out is a game played on graphs. Begin with a graph and a vertex labeling of the graph from the set {0, 1, 2, . . . , ℓ − 1} for ℓ ∈ N. The game is played by toggling vertices: when a vertex is toggled, that vertex and each of its neighbors has its label increased by 1 (modulo ℓ). The game is won when every vertex has label 0. For any n ≥ 2 it is clear that one cannot win the game on K n unless the initial labeling assigns all vertices the same label. Given that K n has the maximum number of edges of any simple graph on n vertices it is natural to ask how many edges can be in a graph so that the Neighborhood Lights Out game is winnable regardless of the initial labeling. We find the maximum number of edges a winnable n-vertex graph can have when at least one of n and ℓ is odd. When n and ℓ are both even we find the maximum size in two additional cases. The proofs of our results require us to introduce a new version of the Lights Out game that can be played given any square matrix.

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“…To determine winnability, we depend heavily on linear algebra methods similar to those in [2,5,9], and [10]. We discuss these methods in Section 2.…”

Section: An Extremal Problem For the Neighborhood Lights Out Game 999mentioning

confidence: 99%

“…The game is won when the zero labeling is achieved. This game was developed independently in [10] and [3] and has been studied in [4,5,9,14], and [6]. The Tiger Electronics Lights Out game is the Neighborhood Lights Out game on a grid graph with ℓ = 2 and has been studied in [1,11], and [15].…”

Section: Introductionmentioning

confidence: 99%

An extremal problem for the neighborhood Lights Out game

Keough

Parker

2024

Discuss. Math. Graph Theory

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“…How the nullity of a graph change when a vertex is removed or what the nullity of the emergent graph becomes when two graphs are joined together by a single or multiple edges was investigated by several authors [3], [10], [8], [4], [5]. However, as far as we know, until now there has been no study which investigates how nullity changes when an edge is added to or removed from a graph.…”

Section: Introductionmentioning

confidence: 99%

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

Batal

2021

Preprint

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Lights out is a game which can be played on any graph G. Initially we have a 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 initial configuration by activating some vertices where each activation switches the state of the vertex and all of its neighbors. If the aim of the game can be accomplished for all initial configurations then G is called always solvable. We call the dimension of the kernel of the closed neighborhood matrix of the graph over the field Z 2 , nullity of G. It turns out that G is always solvable if and only if its nullity is zero. We investigate the problem of how nullity changes when an edge is added to or removed from a graph. As a result we characterize always solvable graphs. We also showed that for every graph with positive nullity there exists an edge whose removal decreases the nullity. Conversely, we showed that for every always solvable graph which is not an even graph with odd order, there exists an edge whose addition increases the nullity.

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“…The connection between the nullities of graphs related to each other by some type of graph operation such as edge/vertex join or removal was first considered by Amin et al [4]. The same subject was investigated by Giffen et al [12], Edwards et al [9] and Ballard et al [5] for a generalized version of the game where (1) is considered over Z k for some integer k ≥ 2. In [4] and [5] the difference ν(G − u) − ν(G), where u is a vertex of graph G, plays an important role in the analysis.…”

Section: Introductionmentioning

confidence: 99%

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

Batal

2020

Preprint

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Lights out is a game that can be played on any simple graph G. A configuration assigns one of the two states on or off to each vertex. For a given configuration, the aim of the game is to turn all vertices 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 numbers of vertices and we prove several characterization results of graphs by using this concept. We show that for every always solvable graph there exists a chain of always solvable subgraphs where each subgraph differs from the preceding one by a vertex. We also characterize always solvable trees by showing that all always solvable trees can be constructed from always solvable subtrees by some special types of connections. 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.

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Lights Out on finite graphs

Cited by 8 publications

References 4 publications

An extremal problem for the neighborhood Lights Out game

An extremal problem for the neighborhood Lights Out game

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

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

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