On realization graphs of degree sequences

Given the degree sequence dd of a graph, the realization graph of dd is the graph having as its vertices the labeled realizations of dd, with two vertices adjacent if one realization may be obtained from the other via an edge-switching operation. We describe a connection between Cartesian products in realization graphs and the canonical decomposition of degree sequences described by R.I. Tyshkevich and others. As applications, we characterize the degree sequences whose realization graphs are triangle-free graphs or hypercubes.