  With sufficient time, double edge-swap Markov chain Monte Carlo (MCMC)
methods are able to sample uniformly at random from many different and
important graph spaces. For instance, for a fixed degree sequence, MCMC methods
can sample any graph from: simple graphs; multigraphs (which may have
multiedges); and pseudographs (which may have multiedges and/or multiple
self-loops). In this note we extend these MCMC methods to `multiloop-graphs',
which allow multiple self-loops but not multiedges and `loopy-multigraphs'
which allow multiedges and single self-loops. We demonstrate that there are
degree sequences on which the standard MCMC methods cannot uniformly sample
multiloop-graphs, and exactly characterize which degree sequences can and
cannot be so sampled. In contrast, we prove that such MCMC methods can sample
all loopy-multigraphs. Taken together with recent work on graphs which allow
single self-loops but no multiedges, this work completes the study of the
connectivity (irreducibility) of double edge-swap Markov chains for all
combinations of allowing self-loops, multiple self-loops and/or multiedges.
Looking toward other possible directions to extend edge swap sampling
techniques, we produce examples of degree and triangle constraints which have
disconnected spaces for all edges swaps on less than or equal to 8 edges.
