The edge-flipping group of a graph

Let X = ( V , E ) be a finite simple connected graph with n vertices and m edges. A configuration is an assignment of one of the two colors, black or white, to each edge of X . A move applied to a configuration is to select a black edge ϵ ∈ E and change the colors of all adjacent edges of ϵ . 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 / 2 Z ) 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.