Preserving Terminal Distances Using Minors

Abstract

We introduce the following notion of compressing an undirected graph $G$ with (nonnegative) edge-lengths and terminal vertices $R\subseteq V(G)$. A distance-preserving minor is a minor $G'$ (of $G$) with possibly different edge-lengths, such that $R\subseteq V(G')$ 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 $f^*(k)$ such that every graph $G$ with $k=|R|$ terminals admits a distance-preserving minor $G'$ with at most $f^*(k)$ vertices? Simple analysis shows that $f^*(k)\le O(k^4)$. Our main result proves that $f^*(k)\ge \Omega(k^2)$, significantly improving on the trivial $f^*(k)\ge k$. 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.