Degree-preserving edge rewiring

Degree-preserving edge rewiring replaces two edges $(u,v)$ and $(x,y)$ by $(u,y)$ and $(x,v)$, preserving every endpoint's degree of a vertex. On simple graphs, reject proposed self-loops and duplicate edges. Symmetric random proposals with rejected moves retained preserve a uniform distribution on the reachable realizations; adequate mixing remains a separate requirement. Connectivity can change unless explicitly constrained.