  Gaussian graphical models are a popular tool to learn the dependence
structure in the form of a graph among variables of interest. Bayesian methods
have gained in popularity in the last two decades due to their ability to
simultaneously learn the covariance and the graph and characterize uncertainty
in the selection. For scalability of the Markov chain Monte Carlo algorithms,
decomposability is commonly imposed on the graph space. A wide variety of
graphical conjugate priors are proposed jointly on the covariance matrix and
the graph with improved algorithms to search along the space of decomposable
graphs, rendering the methods extremely popular in the context of multivariate
dependence modeling. {\it An open problem} in Bayesian decomposable structure
learning is whether the posterior distribution is able to select a meaningful
decomposable graph that it is ``close'' in an appropriate sense to the true
non-decomposable graph, when the dimension of the variables increases with the
sample size. In this article, we explore specific conditions on the true
precision matrix and the graph which results in an affirmative answer to this
question using a commonly used hyper-inverse Wishart prior on the covariance
matrix and a suitable complexity prior on the graph space, both in the
well-specified and misspecified settings. In absence of structural sparsity
assumptions, our strong selection consistency holds in a high dimensional
setting where $p = O(n^{\alpha})$ for $\alpha < 1/3$. We show when the true
graph is non-decomposable, the posterior distribution on the graph concentrates
on a set of graphs that are {\it minimal triangulations} of the true graph.
