A quick proof on the equivalence classes of extended Vogan diagrams

Chuah, Meng Kiat; Hu, Chu Chin

doi:10.1016/j.jalgebra.2006.12.014

Cited by 4 publications

(2 citation statements)

References 7 publications

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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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“…, where I is the identity matrix. The introduction of the lit-only restriction makes the sigma-game harder to analyze and leads to an even richer mathematical structure [6,7,10,12,13,14,15,21,22,24,25,36,37,38,39].…”

Section: Sigma-gamementioning

confidence: 99%

Difference between minimum light numbers of sigma-game and lit-only sigma-game

Wang¹,

Wu²

2009

Preprint

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A 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. The following result is proved in this note: given any starting configuration x of a 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 example to show that the upper bound ℓ + 2 is sharp. Some relevant results and conjectures are also reported.

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Does the lit-only restriction make any difference for the σ-game and σ+-game?

Goldwasser

Wang

2009

European Journal of Combinatorics

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A quick proof on the equivalence classes of extended Vogan diagrams

Cited by 4 publications

References 7 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

Difference between minimum light numbers of sigma-game and lit-only sigma-game

Does the lit-only restriction make any difference for the σ-game and σ+-game?

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