seems to be trivial to solve and therefore mathematically not interesting. Still I think there should be a name and an algorithm how to find it in internet.E.g. at wikipedia. I'd call it the pipeline problem.
I think the solution is to repeatedly select the shortest distance between a connected city and a nonconnected one and connect the two.
Or has someone a counterexample ?
It's not satisfactory for me, though. I want to list the cities in 1-dim and I wand conglomerations to be listed in one group and not possibly scattered. The "subtree grouping problem" ? Or the "province forming problem" ? How to assign the cities from a list to k -to be formed- administrative regions and subregions so the the sum of distances of cities in the same group is minimal.
given a metric space M and an integer k, find disjoint subsets S1..Sk of M whose union is M so to minimize SUM(i=1..k)SUM((x,y) in SixSi) d(x,y)