[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81958-en":3,"doc-seo-81958-105":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},81958,8796095461610,"Oliver","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Deciding Monotonicity of Simple Drawings of the Complete Graph","A drawing of a graph is x-monotone when every vertical line intersects each edge at most once. This work presents an Opn5 q time algorithm to decide whether a simple drawing of the complete graph Kn is weakly isomorphic to an x-monotone drawing, relying on combinatorial representations of the drawing. The same algorithm framework also determines strong isomorphism to x-monotone drawings, establishing efficient recognition for both equivalence notions.","arXiv :2607 .06240v 1 [ cs .CG] 7 Jul 2026  \nDeciding monotonicity of simple drawings of the complete graph  \nOswin Aichholzer∗1, Thomas Hackl†2, Alexander Pilz:2, Gelasio Salazar‡3 ,  \nand Birgit Vogtenhuber§1  \n1 Institute of Algorithms and Theory, Graz University of Technology, Austria  \n2 Institute of Software Technology, Graz University of Technology, Austria  \n3 Instituto de F´ısica, Universidad Aut´onoma de San Luis Potos´ı, Mexico  \nJuly 8, 2026  \nAbstract  \nA drawing of a graph is x-monotone if every vertical line intersects each edge of the graph at most once. We present an Opn5 q time algorithm for deciding whether a simple drawing of the complete graph Kn is weakly isomorphic to an x-monotone drawing. We note that this algorithm can also decide whether a drawing of Kn is strongly isomorphic to an x-monotone drawing.  \n1 Introduction  \nIn a drawing of a graph on some surface, vertices are represented by distinct points and each edge is represented by a Jordan arc whose endpoints are the endvertices of the edge. No edge passes through a vertex, and edges intersect each other in a finite number of points. For simplicity usually it is also assumed that no three edges meet at a common interior point.  \nThroughout this paper we work with simple drawings of the complete graph Kn. We recall that in a simple drawing (also known as a good drawing) in addition to the previous properties no two edges share more than one point (either a common endvertex or a proper crossing), and no edge crosses itself. An important motivation for investigating simple drawings is that every crossingminimal drawing of a graph is simple [18] . The study of simple drawings and their substructures has attracted significant interest in a variety of contexts [2, 3, 7, 9, 13, 15, 17, 19, 20] . We emphasize that throughout this work all drawings under consideration are implicitly assumed to be simple and unless otherwise stated are hosted in the plane R2 .  \nA drawing is x-monotone if every vertical line intersects each edge at most once. See Figure 1 for an illustration. Our main results involve the existence of polynomial time algorithms to test  \n∗ E-mail: [oswin.aichholzer@tugraz.at](oswin.aichholzer@tugraz.at) .  \n†This work was conducted while this author was a postdoctoral researcher at the Institute of Software Technology, Graz University of Technology. He is currently employed in the private sector.  \n‡E-mail: [gelasio.salazar@uaslp.mx](gelasio.salazar@uaslp.mx) .  \n§E-mail: [birgit.vogtenhuber@tugraz.at](birgit.vogtenhuber@tugraz.at) .  \nwhether a given drawing of Kn in the plane is (weakly or strongly) isomorphic to an x-monotone drawing.  \nWe also recall that two drawings D, D1 of the same graph are weakly isomorphic if there is an incidence-preserving bijection between the drawings such that two edges cross in D if and only if their images in D1 cross. Now D and D1 are strongly isomorphic if they induce homeomorphic cell decompositions of the sphere. That is, they are strongly isomorphic if D 1 can be obtained from D by performing an inverse stereographic projection to the sphere, followed by a self-homeomorphism of the sphere, and finally followed by a stereographic projection back to the plane.  \nThere seem to be very few algorithmic results related to x-monotone drawings reported in the literature. Fulek, Pelsmajer, Schaefer, and ˇStefankoviˇc [8] gave an Opn2 q time algorithm that tests whether a graph with given x-coordinates assigned to the vertices has an x-monotone embedding (respecting the given x-coordinates) . Recently, Kynˇcl and Soukup established an NP-hardness result on the related notion of cylindrical monotonicity [14] .  \nOur main result is the following.  \nTheorem 1 . There is an Opn5 q time algorithm that decides if a given drawing of Kn is weakly isomorphic to an x-monotone drawing.  \nWe remark that in Theorem 1 and throughout this paper we do not assume that the input drawings of Kn are given in any particular format.","cbCainPX6poibu21","https://ap.wps.com/l/cbCainPX6poibu21","pdf",434926,7,1,18,"English","en",105,"# Introduction\n## Definitions and motivation\n## Isomorphism notions\n## Prior work and main result\n# Proof of Theorem 1\n## Reduction to lemmas","[{\"question\":\"What does it mean for a graph drawing to be x-monotone?\",\"answer\":\"A drawing is x-monotone if every vertical line intersects each edge at most once. This constrains how edges behave relative to the x-direction.\"},{\"question\":\"What problem does the paper solve algorithmically?\",\"answer\":\"It decides whether a simple drawing of the complete graph Kn is weakly isomorphic to an x-monotone drawing. The method runs in Opn5 q time.\"},{\"question\":\"How are weak and strong isomorphism related in the context of x-monotonicity?\",\"answer\":\"The paper explains that for drawings of Kn in the plane, weak isomorphism to an x-monotone drawing is equivalent to strong isomorphism to an x-monotone drawing, using rotation-system equivalence and properties of triangle mutations that preserve x-monotonicity.\"}]","Deciding Monotonicity of Simple Drawings of the Complete Graph | PDF",1784177289,45,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":88,"head_meta":90,"extra_data":92,"updated_unix":29},"deciding-monotonicity-of-simple-drawings-of-the-complete-graph","",{"@graph":37,"@context":87},[38,55,70],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,52],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":51},"https://docshare.wps.com/document/research-report/",3,{"item":53,"name":13,"@type":44,"position":54},"https://docshare.wps.com/document/deciding-monotonicity-of-simple-drawings-of-the-complete-graph/81958/",4,{"url":53,"name":13,"@type":56,"author":57,"headline":13,"publisher":59,"fileFormat":62,"inLanguage":24,"description":14,"dateModified":63,"datePublished":64,"encodingFormat":62,"isAccessibleForFree":65,"interactionStatistic":66},"DigitalDocument",{"name":9,"@type":58},"Person",{"url":42,"name":60,"@type":61},"DocShare","Organization","application/pdf","2026-08-04","2026-07-16",true,{"@type":67,"interactionType":68,"userInteractionCount":20},"InteractionCounter",{"@type":69},"ViewAction",{"@type":71,"mainEntity":72},"FAQPage",[73,79,83],{"name":74,"@type":75,"acceptedAnswer":76},"What does it mean for a graph drawing to be x-monotone?","Question",{"text":77,"@type":78},"A drawing is x-monotone if every vertical line intersects each edge at most once. This constrains how edges behave relative to the x-direction.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"What problem does the paper solve algorithmically?",{"text":82,"@type":78},"It decides whether a simple drawing of the complete graph Kn is weakly isomorphic to an x-monotone drawing. The method runs in Opn5 q time.",{"name":84,"@type":75,"acceptedAnswer":85},"How are weak and strong isomorphism related in the context of x-monotonicity?",{"text":86,"@type":78},"The paper explains that for drawings of Kn in the plane, weak isomorphism to an x-monotone drawing is equivalent to strong isomorphism to an x-monotone drawing, using rotation-system equivalence and properties of triangle mutations that preserve x-monotonicity.","https://schema.org",{"og:url":53,"og:type":89,"og:title":13,"og:site_name":60,"og:description":14},"article",{"robots":91,"canonical":53},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":94},[95,99,103,107,112,117,121,124,129,132,136],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":96,"show_sort_weight":97,"slug":98},"Story & Novel",90,"story-novel",{"id":48,"doc_module":4,"doc_module_name":47,"category_name":100,"show_sort_weight":101,"slug":102},"Literature",80,"literature",{"id":54,"doc_module":4,"doc_module_name":47,"category_name":104,"show_sort_weight":105,"slug":106},"Exam",70,"exam",{"id":108,"doc_module":4,"doc_module_name":47,"category_name":109,"show_sort_weight":110,"slug":111},5,"Comic",60,"comic",{"id":113,"doc_module":4,"doc_module_name":47,"category_name":114,"show_sort_weight":115,"slug":116},6,"Technology",50,"technology",{"id":20,"doc_module":4,"doc_module_name":47,"category_name":118,"show_sort_weight":119,"slug":120},"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":122,"slug":123},30,"research-report",{"id":125,"doc_module":4,"doc_module_name":47,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":47,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":133,"doc_module":4,"doc_module_name":47,"category_name":134,"show_sort_weight":133,"slug":135},10,"Lifestyle","lifestyle",{"id":137,"doc_module":4,"doc_module_name":47,"category_name":138,"show_sort_weight":108,"slug":139},19,"General","general"]