[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-649824-105":59,"doc-detail-649824-en":129},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":122,"head_meta":124,"extra_data":126,"updated_unix":128},105,"en","crossing-optimal-extension-of-simple-drawings-paper","Crossing-Optimal Extension of Simple Drawings - Paper","","Extension problems in partial graph drawings require completing an input drawing while preserving constraints, with crossing minimization being a central motivation. For plane drawings and related relaxations, tractability and complexity bounds are well studied. In contrast, far fewer results exist for simple drawings, where any two edges intersect at most once. This work proves fixed-parameter tractability for inserting edges into a simple drawing when parameterized by the number of inserted edges and a bound on newly created crossings.",{"@graph":69,"@context":121},[70,84,104],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/crossing-optimal-extension-of-simple-drawings-paper/649824/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":98,"encodingFormat":97,"isAccessibleForFree":99,"interactionStatistic":100},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/crossing-optimal-extension-of-simple-drawings-paper/649824.png","ImageObject",300,407,{"name":92,"@type":93},"SundayMiles","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-24",true,{"@type":101,"interactionType":102,"userInteractionCount":8},"InteractionCounter",{"@type":103},"ViewAction",{"@type":105,"mainEntity":106},"FAQPage",[107,113,117],{"name":108,"@type":109,"acceptedAnswer":110},"What is the crossing-sensitive extension problem studied in the paper?","Question",{"text":111,"@type":112},"Given an incomplete drawing of a graph, the task is to complete it while maintaining drawing properties, with special attention to minimizing crossings.","Answer",{"name":114,"@type":109,"acceptedAnswer":115},"Why are extension problems for simple drawings of particular interest?",{"text":116,"@type":112},"Simple drawings restrict edge intersections so that any pair of edges meets at most once, yet the extension problem has only recently been shown NP-hard even when inserting a single edge.",{"name":118,"@type":109,"acceptedAnswer":119},"What algorithmic results does the paper provide?",{"text":120,"@type":112},"It establishes fixed-parameter tractability for inserting edges into a simple drawing, parameterized by the number of edges to insert and an upper bound on newly created crossings, and also gives a single-exponential fixed-parameter algorithm for inserting exactly one edge.","https://schema.org",{"og:url":83,"og:type":123,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":125,"canonical":83},"index,follow",{"doc_id":127,"site_id":62},649824,1792878339,{"code":4,"msg":5,"data":130},{"doc_id":127,"user_id":131,"nickname":92,"user_avatar":132,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":133,"file_id":134,"file_url":135,"file_type":136,"file_size":137,"view_count":8,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":138,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":144},1099529839202,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","Crossing-Optimal Extension of Simple Drawings  \nRobert Ganian \\#   \nAlgorithms and Complexity Group, TU Wien, Austria Thekla Hamm \\#  \nAlgorithms and Complexity Group, TU Wien, Austria Fabian Klute \\#   \nDeptartment of Information and Computing Sciences, Utrecht University, The Netherlands Irene Parada \\#   \nTU Eindhoven, The Netherlands  \nBirgit Vogtenhuber \\#  Graz University of Technology, Austria  \n~~ Abstract ~~  \nIn extension problems of partial graph drawings one is given an incomplete drawing of an input graph G and is asked to complete the drawing while maintaining certain properties. A prominent area where such problems arise is that of crossing minimization. For plane drawings and various relaxations of these, there is a number of tractability as well as lower-bound results exploring the computational complexity of crossing-sensitive drawing extension problems. In contrast, comparatively few results are known on extension problems for the fundamental and broad class of simple drawings, that is, drawings in which each pair of edges intersects in at most one point. In fact, the extension problem of simple drawings has only recently been shown to be NP-hard even for inserting a single edge.  \nIn this paper we present tractability results for the crossing-sensitive extension problem of simple drawings. In particular, we show that the problem of inserting edges into a simple drawing is fixed-parameter tractable when parameterized by the number of edges to insert and an upper bound on newly created crossings. Using the same proof techniques, we are also able to answer several closely related variants of this problem, among others the extension problem for k-plane drawings. Moreover, using a different approach, we provide a single-exponential fixed-parameter algorithm for the case in which we are only trying to insert a single edge into the drawing.  \n2012 ACM Subject Classification Theory of computation → Parameterized complexity and exact algorithms; Theory of computation → Computational geometry  \nKeywords and phrases Simple drawings, Extension problems, Crossing minimization, FPTalgorithms  \nDigital Object Identifier 10.4230/LIPIcs.ICALP.2021.72 Category Track A: Algorithms, Complexity and Games  \nRelated Version Full Version: [https://arxiv.org/abs/2012.07457](https://arxiv.org/abs/2012.07457) [24]  \nFunding Robert Ganian: Supported by the Austrian Science Fund (FWF) via projects Y1329 ( Parameterized Analysis in Artificial Intelligence ) and P31336 ( New Frontiers for Parameterized Complexity) .  \nThekla Hamm: Supported by the Austrian Science Fund (FWF) via projects Y1329 (Parameterized Analysis in Artificial Intelligence), P31336 (New Frontiers for Parameterized Complexity) and W1255-N23 .  \nFabian Klute: Supported by the Netherlands Organisation for Scientific Research (NWO) under project no. 612.001.651.  \nBirgit Vogtenhuber: Partially supported by Austrian Science Fund (FWF) within the collaborative DACH project Arrangements and Drawings as FWF project I 3340-N35 .  \n© Robert Ganian, Thekla Hamm, Fabian Klute, Irene Parada, and Birgit Vogtenhuber;  \nlicensed under Creative Commons License CC-BY 4.0  \n48th International Colloquium on Automata, Languages, and Programming (ICALP 2021) . Editors: Nikhil Bansal, Emanuela Merelli, and James Worrell; Article No. 72; pp. 72:1–72:17  \nLeibniz International Proceedings in Informatics  \n Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany  \nS  \nC  \nT  \nA  \nE  \n72:2 Crossing-Optimal Extension of Simple Drawings  \nAcknowledgements This work was started during the Austrian Computational Geometry Reunion Meeting in Strobl (Austria), August 10 to 14, 2020 . We thank all the participants for the nice working atmosphere as well as fruitful discussions on this as well as other topics. The authors would also like to thank Eduard Eiben for his insightful comments.  \n 1  Introduction  \nThe study of the crossing number of graphs, that is, the minimum number","cbCaisn7PkUxu1Bd","https://ap.wps.com/l/cbCaisn7PkUxu1Bd","pdf",1071008,17,"English","# Introduction\n## Crossing minimization and drawing extension\n## Simple drawings and extension complexity","[{\"question\":\"What is the crossing-sensitive extension problem studied in the paper?\",\"answer\":\"Given an incomplete drawing of a graph, the task is to complete it while maintaining drawing properties, with special attention to minimizing crossings.\"},{\"question\":\"Why are extension problems for simple drawings of particular interest?\",\"answer\":\"Simple drawings restrict edge intersections so that any pair of edges meets at most once, yet the extension problem has only recently been shown NP-hard even when inserting a single edge.\"},{\"question\":\"What algorithmic results does the paper provide?\",\"answer\":\"It establishes fixed-parameter tractability for inserting edges into a simple drawing, parameterized by the number of edges to insert and an upper bound on newly created crossings, and also gives a single-exponential fixed-parameter algorithm for inserting exactly one edge.\"}]","Crossing-Optimal Extension of Simple Drawings - Paper | PDF",1792808817,43]