[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81856-en":3,"doc-seo-81856-105":31,"detail-sidebar-cat-0-en-105":93},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},81856,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","Paths and Intersections: Minimum Realization of Okamura-Seymour Instances","Investigates the inverse problem for shortest-path metrics arising from Okamura-Seymour (OS) instances. Given an OS metric D on a cyclically ordered terminal set T, the work seeks minimum realizations, meaning disk-embedded graphs with the prescribed terminal order using the fewest edges. It proves that D yields a canonical medial-graph template and that every minimum realization corresponds to the primal graph of an arrangement of this template. It further shows that embedded graph structures can be recovered and edge lengths realizing D computed efficiently.","arXiv :2607 .02883v 1 [ cs .DS] 3 Jul 2026  \nPaths and Intersections: Minimum Realization of Okamura-Seymour  \nInstances  \nYu Chen∗ Pavlo Pylyavskyy† Zihan Tan‡  \nJuly 7, 2026  \nAbstract  \nWe study the inverse problem for shortest-path metrics of Okamura-Seymour (OS) instances. Given an OS metric D on a cyclically ordered terminal set T, the goal is to find minimum realizationsof D, where minimum means having the fewest edges among all disk-embedded realizations with the prescribed terminal order. We show that D determines a canonical medial graph template and every minimum realization is the primal graph of an arrangement of this template. Consequently, the underlying embedded graphs of minimum realizations of D can be recovered, and for each such graph one can efficiently compute edge lengths realizing D.  \nOur algorithm follows a recent approach of analyzing graph structures, by viewing graphs as paths and their intersections, which we believe is of independent interest.  \n∗ National University of Singapore, Singapore. Email: [yu.chen@nus.edu.sg](yu.chen@nus.edu.sg).†University of Minnesota Twin Cities, MN, USA. Email: [ppylyavs@umn.edu](ppylyavs@umn.edu)[ ](ppylyavs@umn.edu)‡University of Minnesota Twin Cities, MN, USA. Email: [ztan@umn.edu](ztan@umn.edu).  \n1 Introduction  \nA typical problem in metric graph theory is the distance realization problem. We are given a metric Don a set T of points (called terminals) and a family G of graphs, and the goal is to decide whether there exists a graph G ∈ G, with the points of T identified with specified vertices of G, such that for every pairs, t ∈ T, distG (s, t) = D (s, t) . As a fundamental problem in metric graph theory, distance realization problems have found various applications in phylogenetics, chemistry, hierarchical classification, and network tomography [Gor87, CGGS01, Dre12, JMNT15] . Notable families of graphs/metrics studied in this problem include ultrametrics, tree, and cactus metrics/graphs [HY65, Per69, Bun74, Dre84, HHMM20] . In this work, we consider another family of graphs: Okamura-Seymour instances.  \nAn Okamura–Seymour (OS) instance (G, T) is a plane graph embedded in a disk with all terminals in T lying on the boundary. OS instances have received much attention in graph algorithms especially in the study of flow-cut gaps [OS81] . Given a metric D and a cyclic order σ on a set T of terminals, the distance realization problem asks if there exists an edge-weighted OS instance (G, T) with terminals appearing in order σ that realizes D as the shortest-path distance metric on T, and the answer is an elegant four-point (sufficient and necessary) condition [HLST88, CO20]: for all quadruples t 1 , t2 , t3 , t4 of terminals appearing in this order,  \nD (t1 , t3 ) + D(t2 , t4 ) ≥ D (t1 , t2 ) + D(t3 , t4 ) . (1)  \nThis condition is often called Kalmanson condition, and the metrics that satisfy it for a fixed cyclic order are called Kalmanson metrics, see [Kal75, BD92, For23, GK24] .  \nThe natural next question after distance realization is minimum realization [FMM+03], where the goal is to find, among all graphs in G realizing a given metric D, the ones with fewest edges. Compared with the distance realization problem which focuses on the “structural property”, the minimum realization problem focuses on the “structural complexity” of a metric, aiming for the minimum-length description of a metric, and therefore falls naturally under the category of graph compression, graph reconstruction, and succinct graph representation.  \nIn this paper, we study the minimum realization problem for the family of Okamura-Seymour instances. We are given a metric D and a cyclic order on a set T of terminals that satisfy all four-point conditions 1 (in this case we say that D is an Okamura-Seymour metric) . The goal is to find the Okamura-Seymour instance (G, T) with minimum number of edges, where terminals lie on the boundary in the given order and (G, T) realizes D as the shortest-p","cbCaipPjY0T2ELqg","https://ap.wps.com/l/cbCaipPjY0T2ELqg","pdf",1457893,6,1,17,"English","en",105,"# Abstract\n# Introduction\n## Distance realization and OS instances\n## Kalmanson condition and OS metrics\n## Minimum realization problem\n## Main theorem and algorithm overview\n## Repelling pairs and forced separation","[{\"question\":\"What is the minimum realization problem for Okamura-Seymour (OS) metrics?\",\"answer\":\"For a given OS metric D with a cyclic order of terminals, it asks for OS disk-embedded instances realizing D as shortest-path distances using the fewest edges possible.\"},{\"question\":\"How does the paper characterize OS metrics from distance data?\",\"answer\":\"It uses the four-point condition for terminals appearing in a fixed cyclic order, known as the Kalmanson condition, to define metrics that correspond to OS instances.\"},{\"question\":\"What does the main theorem guarantee about minimum OS realizations?\",\"answer\":\"It states that the underlying graph structures of all minimum OS realizations can be efficiently recovered, and for each recovered structure, one can efficiently compute nonnegative edge lengths that realize D.\"}]","Paths and Intersections: Minimum Realization of Okamura-Seymour Instances | 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is the minimum realization problem for Okamura-Seymour (OS) metrics?","Question",{"text":77,"@type":78},"For a given OS metric D with a cyclic order of terminals, it asks for OS disk-embedded instances realizing D as shortest-path distances using the fewest edges possible.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"How does the paper characterize OS metrics from distance data?",{"text":82,"@type":78},"It uses the four-point condition for terminals appearing in a fixed cyclic order, known as the Kalmanson condition, to define metrics that correspond to OS instances.",{"name":84,"@type":75,"acceptedAnswer":85},"What does the main theorem guarantee about minimum OS realizations?",{"text":86,"@type":78},"It states that the underlying graph structures of all minimum OS realizations can be efficiently recovered, and for each recovered structure, one can efficiently compute nonnegative edge lengths that realize 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