Preserving Terminal Distances Using Minors

We introduce the following notion of compressing an undirected graph G with (nonnegative) edge-lengths and terminal vertices R subset of V (G). A distance-preserving minor is a minor G' (of G) with possibly different edge-lengths, such that R subset of V(G1) 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* (k) such that every graph G with k = vertical bar R vertical bar terminals admits a distance-preserving minor G' with at most f * (k) vertices? Simple analysis shows that f * (k) < (k4). Our main result proves that f * (k) > Omega (k(2)), significantly improving over the trivial f * (k) >= k. Our lower bound holds even for planar graphs G, in contrast to graphs G of constant treewidth, for which we prove that O(k) vertices suffice.