  This paper proposes a novel representation of decomposable graphs based on
semi-latent tree-dependent bipartite graphs. The novel representation has two
main benefits. First, it enables a form of sub-clustering within maximal
cliques of the graph, adding informational richness to the general use of
decomposable graphs that could be harnessed in applications with behavioural
type of data. Second, it allows for a new node-driven Markov chain Monte Carlo
sampler of decomposable graphs that can easily parallelize and scale. The
proposed sampler also benefits from the computational efficiency of
junction-tree-based samplers of decomposable graphs.
