[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85313-en":3,"doc-seo-85313-105":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},85313,1099514068365,"Aurelia","https://ap-avatar.wpscdn.com/avatar/10000253d8d9f28188e?_k=1776742907772140068",8,"Research & Report","Proximity Measures for Classes of Phylogenetic Networks","Phylogenetic networks model evolutionary histories beyond the limits of single trees, capturing reticulate events such as hybridization and horizontal gene transfer. The work studies restricted classes including tree-child, orchard, and tree-based networks, which formalize different notions of tree-likeness. A central question asks how far a given network is from satisfying a chosen class. Three proximity measures are introduced using leaf addition, valid arc deletion, and arc deletion, with pairwise comparisons, complexity results, and extremal bounds for tree, tree-child, orchard, and tree-based classes.","arXiv :2607 . 11325v1 [math .CO] 13 Jul 2026  \nProximity Measures for Classes of Phylogenetic Networks ∗  \nLeo van Iersel†1, Mark Jones‡1, Esther Julien§1, Yangjing Long¶2, and Yukihiro  \nMurakami ‖ 1  \n1 Delft Institute of Applied Mathematics, TU Delft, the Netherlands  \n2 School of Mathematics and Statistics, Central China Normal University, China  \nJuly 14, 2026  \nAbstract  \nPhylogenetic networks are used to represent the evolutionary history of species. Due to biological interpretations and computational advantages, researchers have focused on restricted classes of phylogenetic networks, such as tree-child, orchard, and tree-based. These classes capture different notions of tree-likeness: tree-child networks require every internal vertex to have a taxon reachable by a tree path, orchard networks are trees with horizontal arcs (for modelling histories rife with horizontal gene transfers), and tree-based networks are trees with additional (not-necessarily horizontal) arcs. A natural question to ask is “how far is a given network from belonging to a particular class?” This motivates the study of proximity measures, which measure the minimum number of graph modifications required to transform a network into one belonging to a particular class. In this paper, we consider three proximity measures based on leaf addition, valid arc deletion, and arc deletion. We study pairwise comparability of the proximity measures, prove complexity results, and derive extremal bounds for the classes of tree, tree-child, orchard, and tree-based networks.  \n1 Introduction  \nPhylogenetic trees are used to represent the evolutionary history of species. While they are effective for illustrating speciation events through vertical descent, they are insufficient in representing more intricate evolutionary processes. Reticulate (net-like) events such as hybridization and horizontal gene transfer (HGT) can give rise to signals that cannot be represented on a single tree [9, 24] . In light of this, phylogenetic networks have received increasing attention due to their capability in elucidating reticulate evolutionary processes.  \nSince the space of all phylogenetic networks is huge and contains extremely complex and possibly unrealistic evolutionary histories, phylogenetic networks are often categorized into different classes based on their topological features. These are often motivated computationally, but some classes are also defined based on their biological relevance [17] . Classical examples of network classes involve the tree-child networks [2] and the tree-based networks [8] . Roughly speaking, tree-child networks are those where every vertex has passed on a gene via vertical descent to an extant species, and tree-based networks are those obtainable from a tree by adding so-called linking arcs between tree arcs. Recent developments have culminated in the introduction of orchard networks, which lie – inclusion-wise – between the two aforementioned network classes [4, 12] . The class has been shown to be both algorithmically attractive and biologically relevant; they are defined as  \n∗ This research was partially funded by the Dutch Research Council (NWO) grant OCENW.KLEIN.125, OCENW.M.21.306, and OCENW.GROOT.2019.015 .  \n†[L.J.J.vanIersel@tudelft.nl](L.J.J.vanIersel@tudelft.nl)[ ](L.J.J.vanIersel@tudelft.nl)‡[m.jones@mdx.ac.uk](m.jones@mdx.ac.uk)  \n§ [E.A.T.Julien@tudelft.nl](E.A.T.Julien@tudelft.nl)[ ](E.A.T.Julien@tudelft.nl)¶[yangjing@ccnu.edu.cn](yangjing@ccnu.edu.cn)[ ](yangjing@ccnu.edu.cn)‖[Y.Murakami@tudelft.nl](Y.Murakami@tudelft.nl)  \nFigure 1: A network on 11 different taxa (excluding the unsampled taxon) of fungi including 5 reticulations, which is part of a larger network from [20] . The directed arcs in the figure are linking arcs, which represent gene transfer highways. In order to make all linking arcs horizontal, we require an additional leaf (unsampled taxon) to represent the evolutionary history.  \nnetworks that can be reduced to ","cbCaic2Apwo1zOvH","https://ap.wps.com/l/cbCaic2Apwo1zOvH","pdf",1540557,1,33,"English","en",105,"# Introduction\n## Phylogenetic networks and classes\n## Proximity measures via graph operations\n# Methods\n## Leaf addition\n## Valid arc deletion\n## Arc deletion","[{\"question\":\"What problem does the paper address about phylogenetic networks?\",\"answer\":\"It quantifies how close a given phylogenetic network is to belonging to specific restricted classes, rather than assuming networks always fall into one class.\"},{\"question\":\"Which proximity measures are studied in the paper?\",\"answer\":\"Three measures are considered based on leaf addition, valid arc deletion, and arc deletion, all defined through the minimum number of graph modifications needed to reach a target class.\"},{\"question\":\"What network classes are used for distance and bounds?\",\"answer\":\"The paper focuses on four classes: trees (no reticulations), tree-child networks, orchard networks, and tree-based 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problem does the paper address about phylogenetic networks?","Question",{"text":75,"@type":76},"It quantifies how close a given phylogenetic network is to belonging to specific restricted classes, rather than assuming networks always fall into one class.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which proximity measures are studied in the paper?",{"text":80,"@type":76},"Three measures are considered based on leaf addition, valid arc deletion, and arc deletion, all defined through the minimum number of graph modifications needed to reach a target class.",{"name":82,"@type":73,"acceptedAnswer":83},"What network classes are used for distance and bounds?",{"text":84,"@type":76},"The paper focuses on four classes: trees (no reticulations), tree-child networks, orchard networks, and tree-based 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