Steiner Point Removal in Graph Metrics

Given a family of graphs F, and graph G ∈ F with weights on the edges, the vertices of G are partitioned into terminals T and Steiner nodes S. The shortest paths (according to edge weights) define a metric on the set of vertices. We wish to embed the set T in a weighted graph G ′ ∈ F such that the distance between any two vertices x, y ∈ T in the graph G ′ is “close” to their distance in G. More precisely, does there exist a graph G ′ on the set T, such that for every x, y ∈ T, dG(x, y) ≤ dG′(x, y) ≤ αdG(x, y). We obtain results for the family of outerplanar graphs. We show that we can remove Steiner nodes from any outerplanar graph G and embed the terminals in another outerplanar graph G′ with constant α. Moreover, in our algorithm, G ′ is a minor of G. This strictly improves the class of graphs for which Steiner point removal can be done with constant distortion. The previously best known result was for trees due to Gupta [6].