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Bayesian inference for undirected graphical models is mostly restricted tothe class of decomposable graphs, as they enjoy a rich set of properties makingthem amenable to high-dimensional problems. While parameter inference isstraightforward in this setup, inferring the underlying graph is a challengedriven by the computational difficulty in exploring the space of decomposablegraphs. This work makes two contributions to address this problem. First, weprovide sufficient and necessary conditions for when multi-edge perturbationsmaintain decomposability of the graph. Using these, we characterize a simpleclass of partitions that efficiently classify all edge perturbations by whetherthey maintain decomposability. Second, we propose a novel parallelnon-reversible Markov chain Monte Carlo sampler for distributions over junctiontree representations of the graph. At every step, the parallel sampler executessimultaneously all edge perturbations within a partition. Through simulations,we demonstrate the efficiency of our new edge perturbation conditions and classof partitions. We find that our parallel sampler yields improved mixingproperties in comparison to the single-move variate, and outperforms currentstate-of-the-arts methods in terms of accuracy and computational efficiency.The implementation of our work is available in the Python package parallelDG.

Parallel sampling of decomposable graphs using Markov chain on junction trees