Rewiring of two disjoint edges is a natural operation on a graph G that preserves the degree sequence.
Let (u,v) and (x,y) be two disjoint edges with 4 distinct vertices. In a rewiring of these edges we omit both and replace them with the edges (u,x) and (v,y) or with the edges (u,y) and (v,x) if these edges do not already exist.
Depending on the existing edges from u and v to x and y, sometimes there are 2 possible ways to do the rewiring, sometimes only one way, and sometimes it is impossible.