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 two colors, black or white, to each edge of X. A move applied to a configuration is to select a black edge 2 ∈ E and change the colors of all adjacent edges of 2. 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 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.