The flipping puzzle on a graph

Huang, Hau-Wen; Weng, Chih-wen

doi:10.1016/j.ejc.2009.08.001

Cited by 4 publications

(2 citation statements)

References 15 publications

(14 reference statements)

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“…The lit-only σ-game and its closely related variants have been studied not only for fun by amateurs [40] but are also studied by mathematicians for mathematical fun [8,10,25,26,44,45,46,48] and from the perspectives of error-correcting codes and combinatorial game theory [12,13,14,15,17], Lie algebras and Coxeter groups [4,5,6,7,29,30,31,32,39], statistical physics of social balance [33,34], and general reachability analysis [27]. The study of the σ-game has a longer history than that of the lit-only σ-game and is still mushrooming; see [1,2,3,9,10,11,16,18,19,20,21,22,23,24,25,28,35,36,40,41,42,…”

Section: Definitions and Backgroundmentioning

confidence: 99%

Minimum Light Numbers in the $\sigma$-Game and Lit-Only $\sigma$-Game on Unicyclic and Grid Graphs

Goldwasser¹,

Wang²,

Wu³

2011

Electron. J. Combin.

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Consider a graph each of whose vertices is either in the ON state or in the OFF state and call the resulting ordered bipartition into ON vertices and OFF vertices a configuration of the graph. A regular move at a vertex changes the states of the neighbors of that vertex and hence sends the current configuration to another one. A valid move is a regular move at an ON vertex. For any graph $G,$ let $\mathcal{D}(G)$ be the minimum integer such that given any starting configuration $\bf x$ of $G$ there must exist a sequence of valid moves which takes $\bf x$ to a configuration with at most $\ell +\mathcal{D}(G)$ ON vertices provided there is a sequence of regular moves which brings $\bf x$ to a configuration in which there are $\ell$ ON vertices. The shadow graph $\mathcal{S}(G)$ of a graph $G$ is obtained from $G$ by deleting all loops. We prove that $\mathcal{D}(G)\leq 3$ if $\mathcal{S}(G)$ is unicyclic and give an example to show that the bound $3$ is tight. We also prove that $\mathcal{D}(G)\leq 2$ if $ G $ is a two-dimensional grid graph and $\mathcal{D}(G)=0$ if $\mathcal{S}(G)$ is a two-dimensional grid graph but not a path and $G\neq \mathcal{S}(G)$.

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Section: Definitions and Backgroundmentioning

confidence: 99%

Minimum Light Numbers in the $\sigma$-Game and Lit-Only $\sigma$-Game on Unicyclic and Grid Graphs

Goldwasser¹,

Wang²,

Wu³

2011

Electron. J. Combin.

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“…Finally, as hinted by Eqs. (1) and (2), the lit-only σ -game is naturally associated with certain groups generated by transvections and hence is also studied (implicitly) in some algebra settings [8,9].…”

Section: σ-Game and Lit-only σ-Gamementioning

confidence: 99%

Lit-only σ-game on pseudo-trees

Wang

2010

Discrete Applied Mathematics

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a b s t r a c tA configuration of a graph is an assignment of one of two states, ON or OFF, to each vertex of it. A regular move at a vertex changes the states of the neighbors of that vertex. A valid move is a regular move at an ON vertex. A pseudo-tree is a graph obtained from a tree by attaching zero or more loops. The following result is proved in this note: given any starting configuration x of a pseudo-tree, if there is a sequence of regular moves which brings x to another configuration in which there are ℓ ON vertices then there must exist a sequence of valid moves which takes x to a configuration with at most ℓ+2 ON vertices. We provide an example to show that the upper bound ℓ + 2 is sharp. Some related problems and conjectures are also reported.

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The edge-flipping group of a graph

Huang

Weng

2010

European Journal of Combinatorics

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Let X = (V, E) be a finite simple connected graph with n vertices and m edges. A configuration is an assignment of one of two colors, black or white, to each edge of X. A move applied to a configuration is to select a black edge ǫ ∈ E and change the colors of all adjacent edges of ǫ. Given an initial configuration and a final configuration, try to find a sequence of moves that transforms the initial configuration into the final configuration. This is the edge-flipping puzzle on X, and it corresponds to a group action. This group is called the edge-flipping group W E (X) of X. This paper shows that if X has at least three vertices, W E (X) is isomorphic to a semidirect product of (Z/2Z) k and the symmetric group S n of degree n, where k = (n − 1)(m − n + 1) if n is odd, k = (n − 2)(m − n + 1) if n is even, and Z is the additive group of integers.An ordered pair X = (V, E) is a finite simple graph if V is a finite set and E is a set of some 2-element subsets of V. The elements of V are called vertices of X and the elements of E are called edges of X. Two vertices

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The flipping puzzle on a graph

Cited by 4 publications

References 15 publications

Minimum Light Numbers in the $\sigma$-Game and Lit-Only $\sigma$-Game on Unicyclic and Grid Graphs

Minimum Light Numbers in the $\sigma$-Game and Lit-Only $\sigma$-Game on Unicyclic and Grid Graphs

Lit-only σ-game on pseudo-trees

The edge-flipping group of a graph

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