Is there a name for the problem of spatially organizing a graph as to minimize total edge length? The problem is that of spatially (with or without a fixed spatial dimension) organizing a graph so that each node becomes a cell in a grid, and each edge becomes a line, such that the total combined length of all lines (edges) is the minimal.

I want to know if that problem has a name, and how complex it is. I also would like to know if this problem is easier if done incrementally (i.e., starting with an empty graph and adding/removing nodes as to maintain that property).
This is a version of graph drawing, restricted to vertices being on a grid, but allowing more than two dimensions.  The one-dimensional case is an example of a graph layout problem, known as Minimum Linear Arrangement. In all of these problems, one is interested in minimizing some metric of the resulting diagram, which could be the number of edge crossings or, in your case, the total length of the edges.

These kinds of problem are very often NP-hard, even in the one-dimensional case: in particular Minimum Linear Arrangement, the 1D version of your problem, is NP-hard. There's a survey of graph layout problems by D&iacute;az, Petit and Serna (ACM Surveys, 34:313&ndash;356, 2002; .ps.gz).