Proposition: The shortened metric

Define

This is the shortest-path metric obtained by adjoining an undirected edge of length $r$ between $P,Q$. It is integer valued, has finite balls, and all repartitioned blocks have $d_W$-diameter at most $r$.

Proof. Represent the original metric by the complete weighted graph with edge lengths $d(x,y)$. The triangle inequality allows every path not using the added edge to be shortened to one original edge. A shortest path uses the new edge at most once, since a repeated use contains a positive-length loop that can be removed. This gives the displayed formula and proves that it is a metric.

A $d_W$-ball is contained in a union of three finite $d$-balls: one about its center and ones about $P,Q$, with radius no larger than the specified ball radius. Finally $d_W\le d$ and $d_W(P,Q)\le r$.