Paths and Intersections: Repelling Pairs

Abstract

We study two inverse problems for shortest-path metrics of Okamura-Seymour instances: recognizing metrics realizable by outerplanar graphs, and computing minimum-edge Okamura-Seymour realizations. We introduce the notion of \emph{repelling pairs}, a metric certificate that the shortest paths corresponding to two terminal pairs must be vertex-disjoint in every realization. Our central structural result is that, for an Okamura-Seymour metric with a prescribed cyclic order, a terminal path structure in an Okamura-Seymour instance can be realized by nonnegative edge lengths if and only if the paths assigned to every repelling pair are vertex-disjoint. Building on the notion of repelling pairs, we give algorithmic answers to the inverse problems. First, we design an algorithm that, given a metric, decides in polynomial time whether or it admits an outerplanar realization and constructs one when one exists. Second, given an Okamura-Seymour metric, we efficiently compute a canonical medial template whose crossing number equals the minimum number of edges in any Okamura-Seymour realization. The minimum-edge graph structures are exactly the primal graphs of arrangements of this template, and each can be assigned realizing edge lengths in polynomial time.