We consider the problem of selecting a subset of the dimensions of an image manifold that best preserves the underlying local structure in the original data. We have previously shown that masks which preserve the data neighborhood graph are well suited to global manifold learning algorithms. However, local manifold learning algorithms leverage a geometric structure beyond that captured by this neighborhood graph. In this paper, we present a mask selection algorithm that further preserves this additional structure by designing an extended data neighborhood graph that connects all neighbors of each data point, forming local cliques. Numerical experiments show the improvements achieved by employing the extended graph in the mask selection process.