Neighborhood graph construction is usually the first step in algorithms for isometric data embedding and manifold learning that cope with the problem of projecting high dimensional data to a low space. This paper begins by explaining the algorithmic fundamentals of techniques for isometric data embedding and derives a general classification of these techniques. We will see that the nearest neighbor approaches commonly used to construct neighborhood graphs do not guarantee connectedness of the constructed neighborhood graphs and, consequently, may cause an algorithm fail to project data to a single low dimensional coordinate system. In this paper, we review three existing methods to construct k-edge-connected neighborhood graphs and propose a new method to construct k-connected neighborhood graphs. These methods are applicable to a wide range of data including data distributed among clusters. Their features are discussed and compared through experiments.
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