Preserving terminal distances using minors

We introduce the following notion of compressing an undirected graph G with (nonnegative) edge-lengths and terminal vertices R ⊆ V (G). A distance-preserving minor is a minor G′ (of G) with possibly different edge-lengths, such that R ⊆ V (G′) and the shortest-path distance between every pair of terminals is exactly the same in G and in G′. We ask: what is the smallest f*(κ) such that every graph G with κ = |R| terminals admits a distance-preserving minor G with at most f*(κ) vertices? Simple analysis shows that f*(κ) ≤ O(κ⁴). Our main result proves that f*(κ) ≥ σ(κ²), significantly improving on the trivial f*(κ) ≥ κ. Our lower bound holds even for planar graphs G, in contrast to graphs G of constant treewidth, for which we prove that O(κ) vertices suffice.