That is not a uniqe property of Greville points. Locations of control points can be computed after moving an edit point which is a member of a different set of edit points. There are limitations on the locations of edit points with a minimum requirement that at least one edit point be within the region of influence of each control point, and a additional restriction if a banded matrix is desired. While Greville points are a good choice for edit points they are not the only choice. The edit points do not need to be Greville points to have a solvable banded matrix. I’ve written code it the distant past which for arbitrary sets of edit points.
Are there any other properties of Greville points which make them desirable to use as edit points? For instance does the use of Greville points allow the use of a special algorithm for solving for the new control point locations? Or are Greville points used because they lead to well behaved matrices and and their locations are easy to calculate? I was hoping someone might be able to provide a reference which discussed Greville points as edit points.