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With sufficient time, double edge-swap Markov chain Monte Carlo (MCMC)methods are able to sample uniformly at random from many different andimportant graph spaces. For instance, for a fixed degree sequence, MCMC methodscan sample any graph from: simple graphs; multigraphs (which may havemultiedges); and pseudographs (which may have multiedges and/or multipleself-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 aredegree sequences on which the standard MCMC methods cannot uniformly samplemultiloop-graphs, and exactly characterize which degree sequences can andcannot be so sampled. In contrast, we prove that such MCMC methods can sampleall loopy-multigraphs. Taken together with recent work on graphs which allowsingle self-loops but no multiedges, this work completes the study of theconnectivity (irreducibility) of double edge-swap Markov chains for allcombinations of allowing self-loops, multiple self-loops and/or multiedges.Looking toward other possible directions to extend edge swap samplingtechniques, we produce examples of degree and triangle constraints which havedisconnected spaces for all edges swaps on less than or equal to 8 edges.

Swap connectivity for two graph spaces between simple and pseudo graphs and disconnectivity for triangle constraints