Finding near neighbors

Question

I need to find "near" neighbors among a set of points.

There are 10 points in the above image. Red lines are edges from the Delaunay Triangulation, black stars mark the mid-lines of the edges, blue lines are the Voronoi tesselation. Point 1 has three "near" neighbors, i.e. 4, 6, and 7, but not 2 and 3, who are almost in line with the edge 1-7, but much further away.

What is a good way to identify the near neighbors (or "good" edges)? Looking at the figure, it seems to me that either selecting edges whose mid-point falls onto the intersection with the Voronoi lines, or considering as "near" neighbors those with touching Voronoi cells could be a good solution (the classification of 3-5 can go either way). Is there an efficient way of implementing either of the solutions in Matlab (I'd be happy to get a good general algorithm that I can then translate to Matlab, btw)?