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.
Related
BibTeX
@misc{chen2026paths,
title = {{Paths and Intersections: Repelling Pairs}},
author = {Yu Chen and Pavlo Pylyavskyy and Zihan Tan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.02883},
eprint = {2607.02883},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.02883},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Paths and Intersections: Repelling Pairs
AU - Yu Chen
AU - Pavlo Pylyavskyy
AU - Zihan Tan
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2607.02883
UR - https://arxiv.org/abs/2607.02883
AB - 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.
ER -