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The flipping group of a line graph
, 2008
"... Let X be a simple connected graph with n vertices and m edges. Every vertex of X is assigned either black state or white state. We move by selecting a vertex v of X having black state and then change the states of all neighbors of v. This is the flipping puzzle on X and it corresponds to a group act ..."
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Let X be a simple connected graph with n vertices and m edges. Every vertex of X is assigned either black state or white state. We move by selecting a vertex v of X having black state and then change the states of all neighbors of v. This is the flipping puzzle on X and it corresponds to a group action. We referred this group to the flipping group of X. In this paper, we are mainly concerned about the flipping group on the line graph L(X) of X. We show that the flipping group of L(X) is isomorphic to a semidirect product of (Z/2Z) k and the symmetric group Sn, 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.
The flipping puzzle on a graph
, 2008
"... Let S be a connected graph which contains an induced path of n −1 vertices, where n is the order of S. We consider a puzzle on S. A configuration of the puzzle is simply an ndimensional column vector over {0,1} with coordinates of the vector indexed by the vertex set S. For each configuration u wit ..."
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Let S be a connected graph which contains an induced path of n −1 vertices, where n is the order of S. We consider a puzzle on S. A configuration of the puzzle is simply an ndimensional column vector over {0,1} with coordinates of the vector indexed by the vertex set S. For each configuration u with a coordinate us = 1, there exists a move that sends u to the new configuration which flips the entries of the coordinates adjacent to s in u. We completely determine if one configuration can move to another in a sequence of finite steps.
The edgeflipping group of a graph∗ Hauwen Huang † Chihwen Weng‡
, 2009
"... 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 ∈ E and change the colors of all adjacent edges of . Given an initial ..."
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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 ∈ E and change the colors of all adjacent edges of . 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 edgeflipping puzzle on X, and it corresponds to a group action. This group is called the edgeflipping group WE(X) of X. This paper shows that if X has at least three vertices, WE(X) is isomorphic to a semidirect product of (Z/2Z)k and the symmetric group Sn 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.