Eliminating Steiner Vertices in Graph Metrics

Given an edge-weighted undirected graph G and a subset of “required” vertices R ⊆ V (G), called the terminals, we want to find a minor G′ with possibly different edge-weights, that retains distances between all terminal-pairs exactly, and is as small as possible. We prove that every graph G with n vertices and k terminals can be reduced (in this sense) to a minor G′ with O(k4) vertices and edges. We also give a lower bound of Ω(k2) on the number of vertices required. The O(k4) upper bound on the size of the minor is achieved using a specific construction for minors, which we call Oriented Minors. For this specific method we show that the upper bound is tight. The Ω(k2) lower bound is proved by an even stronger claim; there are planar graphs G such that any planar graph that preserves distances between terminals in G has Ω(k2) vertices. When restricting the graphs G and G′ to trees, we prove that 2k− 2 vertices are sufficient and necessary. Another version of this problem requires that V (G′) = R and asks for G′ that approximates the distances between terminals within a constant factor. Previous results proved that this is possible in trees and in outerplanar graphs, and termed this problem Steiner Point Removal. We study a particular planar graph G that we suspected would give a super-constant lower bound on the approximation factor. We refute this suspicion, finding an outerplanar minor of G achieving constant approximation. An interesting generalization of this result is that for any distance metric on the terminals {0, 1, .., k} adhering to a certain monotonicity rule, there exists an outerplanar graph that approximates the metric within a constant factor.