  `Double edge swaps' transform one graph into another while preserving the
graph's degree sequence, and have thus been used in a number of popular Markov
chain Monte Carlo (MCMC) sampling techniques. However, while double edge-swaps
can transform, for any fixed degree sequence, any two graphs inside the classes
of simple graphs, multigraphs, and pseudographs, this is not true for graphs
which allow self-loops but not multiedges (loopy graphs). Indeed, we exactly
characterize the degree sequences where double edge swaps cannot reach every
valid loopy graph and develop an efficient algorithm to determine such degree
sequences. The same classification scheme to characterize degree sequences can
be used to prove that, for all degree sequences, loopy graphs are connected by
a combination of double and triple edge swaps. Thus, we contribute the first
MCMC sampler that uniformly samples loopy graphs with any given sequence.
