If it is constrained to bury the cable only along certain paths (e.g.
roads), then there would be a graph containing the points (e.g. Some of the paths might be more expensive, because they are longer, or require the cable to be buried deeper; these paths would be represented by edges with larger weights.
We describe an efficient algorithm for maintaining a minimum spanning tree (MST) in a graph subject to a sequence of edge weight modifications.
The sequence of minimum spanning trees is computed offline, after the sequence of modifications is known.
A fourth algorithm, not as commonly used, is the reverse-delete algorithm, which is the reverse of Kruskal's algorithm. Several researchers have tried to find more computationally-efficient algorithms.