发表机构
National University of Singapore; University of Minnesota Twin Cities（新加坡国立大学; 明尼苏达大学双城分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究冈村-西摩实例最短路径度量反问题，给定循环有序终端集上的OS度量D，目标是找到其最小实现，通过分析图结构，证明D确定规范中间图模板，可恢复最小实现的底层嵌入图并计算边长。
AI 中文摘要
我们研究冈村 - 西摩（OS）实例的最短路径度量的逆问题。给定循环有序终端集\(T\)上的OS度量\(D\)，目标是找到\(D\)的最小实现，其中最小是指在具有规定终端顺序的所有圆盘嵌入实现中边数最少。我们表明\(D\)确定一个规范的中间图模板，并且每个最小实现是该模板排列的原始图。因此，可以恢复\(D\)的最小实现的底层嵌入图，并且对于每个这样的图，可以有效地计算实现\(D\)的边长。我们的算法遵循最近通过将图视为路径及其交点来分析图结构的方法，我们认为这具有独立的研究价值。
英文摘要
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.
CommentsThis version merges the previous version of this paper with arXiv:2606.25827
