[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83353-en":3,"doc-seo-83353-105":29,"detail-sidebar-cat-0-en-105":91},{"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},83353,687197207919,"Theodora","https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552",8,"Research & Report","The ⊖-metric to compare phylogenetic networks","We introduce two operational distances for comparing rooted phylogenetic networks using the ⊖-operator, which removes a vertex while preserving ancestor relations among remaining vertices. The distance 􀀳 ⊖ counts the minimum number of such removals to obtain isomorphic networks, while 􀀳 ignores shortcut arcs and compares induced ancestry structures. Both yield metric properties under appropriate isomorphism notions. They extend the Robinson–Foulds distance on trees and are bounded below by hardwired cluster distances, with polynomial-time computability on several network classes and NP-hardness results elsewhere.","arXiv :2607 .08259v 1 [ cs .DM] 9 Jul 2026  \nThe ⊖-metric to compare phylogenetic networks  \nMarc Hellmuth 1,2 , Manuel Lafond3 , and Guillaume E. Scholz4  \n1 Department of Computer Science, Leipzig University, DE-04109 Leipzig, Germany  \n2 Department of Mathematics, Faculty of Science, Stockholm University, SE-10691 Stockholm, Sweden  \n3 Universit de Sherbrooke, Canada  \n4 Institute of Mathematics and Computer Science, Greifswald University, DE-17849 Greifswald, Germany  \nAbstract  \nWe introduce two novel distances for comparing rooted phylogenetic networks based on the ⊖-operator, which removes a vertex while preserving the ancestor relations among the remaining vertices. The distance 􀀳 ⊖ measures the minimum number of such removals needed to obtain isomorphic networks, whereas 􀀳 ignores shortcut arcs and therefore compares the induced ancestry structures. We show that 􀀳 ⊖ is a metric up to leaffixing isomorphism and that 􀀳 is a metric up to shortcut-free isomorphism. Moreover, both distances extend the Robinson–Foulds distance on phylogenetic trees and are bounded below by the hardwired cluster distances. For several broad network classes, including tree-child, normal, level-1, and regular networks, 􀀳 can be computed in polynomial time. In contrast, computing 􀀳 ⊖ is NP-hard, W[2]-hard when parameterized by the distance value, and admits no polynomial-time constant-factor approximation unless P = NP. Although computing 􀀳 is NPhard in general, for distinct-cluster networks it reduces to Vertex Cover, yielding a fixed-parameter algorithm and a polynomial-time 2-approximation.  \nKeywords: phylogenetic networks; network comparison; operational distance; Robinson-Foulds distance; hardwired clusters; shortcut-free networks; distinct-cluster networks; vertex cover; fixed-parameter tractability; approximation algorithms  \n1 Introduction  \nReconstructing evolutionary histories is a central task in computational biology. Phylogenetic trees provide a suitable model when evolution proceeds exclusively through branching events. However, processes such as hybridization, horizontal gene transfer, and recombination may produce reticulate patterns of ancestry that cannot be represented faithfully by a tree. Rooted phylogenetic networks provide a more general framework for describing such evolutionary histories, and a growing number of methods are available for reconstructing them from biological data [32, 36, 44, 47] .  \nThe increasing number of reconstruction methods creates a corresponding need for meaningful ways to compare their outputs. For example, one may wish to compare an inferred network with a simulated or otherwise known reference network, compare the results produced by different reconstruction methods, or quantify the variability among networks obtained from different data sets. This is generally more difficult for networks than for trees. In a phylogenetic tree, features such as clusters, displayed triplets, and ancestor relations are tightly linked and determine the tree topology uniquely [42] . For phylogenetic networks, however, these features capture different aspects of the structure, and distinct networks may agree on some of them while differing on others [17, 23, 49] . Consequently, there is no single universally accepted distance for comparing arbitrary phylogenetic networks [27, 46] .  \nSeveral dissimilarity measures have been proposed. One of the most widely used is the hardwired cluster distance, which counts the clusters that occur in one network but not in the other [7] . This generalizes the classical Robinson-Foulds distance on phylogenetic trees [41] . For general networks, however, the hardwired cluster distance is only a pseudometric: two non-isomorphic networks may induce the same cluster set and therefore have distance  \nT N ⊖ {w} (N ⊖ {w}) − N ⊖ {u,w} = T ⊖ {u}  \na b c d a b c d a b c d a b c d a b c d  \nFigure 1: A network 􀀣, a phylogenetic tree 􀀩, and several ⊖-reductions. The arc 􀁤 → 􀀲 in 􀀣 ⊖ 􀁆 is a sh","cbCaisSx22omyJP4","https://ap.wps.com/l/cbCaisSx22omyJP4","pdf",672546,1,27,"English","en",105,"# Introduction\n## Need for distance measures\n## Existing feature-based dissimilarities\n## Operational distances and computational complexity","[{\"question\":\"What is the ⊖-operator used in the paper’s distance definitions?\",\"answer\":\"The ⊖-operator removes a vertex while preserving the ancestor relations among the remaining vertices in a rooted phylogenetic network.\"},{\"question\":\"How do the two proposed distances differ in what they consider?\",\"answer\":\"The distance 􀀳 ⊖ measures the minimum number of ⊖-removals needed to reach isomorphic networks, whereas 􀀳 ignores shortcut arcs and instead compares induced ancestry structures.\"},{\"question\":\"What are the main computational complexity results for computing these distances?\",\"answer\":\"Computing 􀀳 ⊖ is NP-hard and W[2]-hard parameterized by the distance value, with no polynomial-time constant-factor approximation unless P = NP. For 􀀳, general computation is NP-hard, but for distinct-cluster networks it reduces to Vertex Cover, enabling a fixed-parameter algorithm and a polynomial-time 2-approximation.\"}]",1784186949,68,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"the-metric-to-compare-phylogenetic-networks","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/the-metric-to-compare-phylogenetic-networks/83353/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the ⊖-operator used in the paper’s distance definitions?","Question",{"text":75,"@type":76},"The ⊖-operator removes a vertex while preserving the ancestor relations among the remaining vertices in a rooted phylogenetic network.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do the two proposed distances differ in what they consider?",{"text":80,"@type":76},"The distance 􀀳 ⊖ measures the minimum number of ⊖-removals needed to reach isomorphic networks, whereas 􀀳 ignores shortcut arcs and instead compares induced ancestry structures.",{"name":82,"@type":73,"acceptedAnswer":83},"What are the main computational complexity results for computing these distances?",{"text":84,"@type":76},"Computing 􀀳 ⊖ is NP-hard and W[2]-hard parameterized by the distance value, with no polynomial-time constant-factor approximation unless P = NP. For 􀀳, general computation is NP-hard, but for distinct-cluster networks it reduces to Vertex Cover, enabling a fixed-parameter algorithm and a polynomial-time 2-approximation.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]