Resolving the Steiner Point Removal Problem in Planar Graphs via Shortcut Partitions

TL;DR

This paper uses shortcut partitions of planar graphs to solve the Steiner point removal problem, creating a minor that preserves terminal distances within a constant factor, thus answering a longstanding open question.

Contribution

The authors apply shortcut partitions to resolve the Steiner point removal problem in planar graphs, establishing a new method for distance preservation among terminals.

Findings

Constructed a minor preserving terminal distances within a constant factor

Resolved an open problem in Steiner point removal for planar graphs

Introduced a novel application of shortcut partitions

Abstract

Recently the authors [CCLMST23] introduced the notion of shortcut partition of planar graphs and obtained several results from the partition, including a tree cover with $O(1)$ trees for planar metrics and an additive embedding into small treewidth graphs. In this note, we apply the same partition to resolve the Steiner point removal (SPR) problem in planar graphs: Given any set $K$ of terminals in an arbitrary edge-weighted planar graph $G$, we construct a minor $M$ of $G$ whose vertex set is $K$, which preserves the shortest-path distances between all pairs of terminals in $G$ up to a constant factor. This resolves in the affirmative an open problem that has been asked repeatedly in literature.