Locally linear embedding is a nonlinear method for dimensionality reduction and manifold learning. It requires well-sampled input data in high dimensional space so that neighborhoods of all data points overlap with each other. In this paper, we build connected neighborhood graphs for the purpose of assigning neighbor points. A few methods are examined to build connected neighborhood graphs. They have made LLE applicable to a wide range of data including under-sampled data and non-uniformly distributed data. These methods are compared through experiments on both synthetic and real world data sets.