Lights Out for graphs related to one another by constructions

Ballard, Laura E.; Budge, Erica; Stephenson, Darin R.

doi:10.2140/involve.2019.12.181

Cited by 9 publications

(9 citation statements)

References 8 publications

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“…Variants of Lights Out with more than one toggle mode have been considered by many authors. [4] [22] It is possible to extend our results in this direction.…”

Section: Future Workmentioning

confidence: 59%

Lights Out On A Random Graph

Forrest¹,

Nicole²

2021

Preprint

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We consider the generalized game Lights Out played on a graph and investigate the following question: for a given positive integer n, what is the probability that a graph chosen uniformly at random from the set of graphs with n vertices yields a universally solvable game of Lights Out? When n ≤ 11, we compute this probability exactly by determining if the game is universally solvable for each graph with n vertices. We approximate this probability for each positive integer n with n ≤ 87 by applying a Monte Carlo method using 1,000,000 trials. We also perform the analogous computations for connected graphs.

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“…Variants of Lights Out with more than one toggle mode have been considered by many authors. [4] [22] It is possible to extend our results in this direction.…”

Section: Future Workmentioning

confidence: 59%

Lights Out On A Random Graph

Forrest¹,

Nicole²

2021

Preprint

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“…The Colored Lights Out game on a particular class of graphs, so called subdivided caterpillars, has been treated in [21]. In [2] harmonic functions on graphs are used to formulate the Colored Lights out game. In [17] they observed the number of equivalence classes of this problem.…”

Section: Proofmentioning

confidence: 99%

Lights Out on graphs

2021

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We model the Lights Out game on general simple graphs in the framework of linear algebra over the field $$\mathbb{F}_{2}$$ F 2 . Based upon a version of the Fredholm alternative, we introduce a separating invariant of the game, i.e., an initial state can be transformed into a final state if and only if the values of the invariant of both states agree. We also investigate certain states with particularly interesting properties. Apart from the classical version of the game, we propose several variants, in particular a version with more than only two states (light on, light off), where the analysis relies on systems of linear equations over the ring $$\mathbb{Z}_{n}$$ Z n . Although it is easy to find a concrete solution of the Lights Out problem, we show that it is NP-hard to find a minimal solution. We also propose electric circuit diagrams to actually realize the Lights Out game.

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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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Lights Out for graphs related to one another by constructions

Cited by 9 publications

References 8 publications

Lights Out On A Random Graph

Lights Out On A Random Graph

Lights Out on graphs

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

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