  We describe the implementation of the Giudici-Green Metropolis sampling
method for decomposable graphs using a variety of structures to represent the
graph. These comprise the graph itself, the Junction tree, the Almond tree and
the Ibarra clique-separator graph. For each structure, we describe the process
for ascertaining whether adding or deleting a specific edge results in a new
graph that is also decomposable, and the updates that need to be made to the
structure if the edge perturbation is made. For the Almond tree and Ibarra
graph these procedures are novel. We find that using the graph itself is
generally at least competitive in terms of computational efficiency for a
variety of graph distributions, but note that the other structures may allow
and suggest samplers using different perturbations with lower rejection rates
and/or better mixing properties. The sampler has applications in estimating
graphical models for systems of multivariate Gaussian or Multinomial variables.
