[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-665055-105":59,"doc-detail-665055-en":130},{"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":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","beyond-degree-four-near-orthogonal-planar-drawings","Beyond Degree Four - Near-Orthogonal Planar Drawings","","Orthogonal planar drawings offer clear, axis-aligned structure in graph drawing, but they exist only for planar graphs whose maximum degree is at most four. This work studies planar polyline drawings for graphs with unrestricted maximum degree, defining “near-orthogonal” via the number of non-orthogonal faces. Even for triconnected graphs with unique planar embedding, deciding existence with at most h non-orthogonal faces is NP-complete. The paper then develops parameterized and approximation results, including linear-time FPT for fixed embeddings and algorithms based on treewidth and related parameters.",{"@graph":69,"@context":122},[70,84,105],{"@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/beyond-degree-four-near-orthogonal-planar-drawings/665055/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/beyond-degree-four-near-orthogonal-planar-drawings/665055.png","ImageObject",300,407,{"name":92,"@type":93},"fresh2021","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-29",true,{"@type":102,"interactionType":103,"userInteractionCount":19},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What defines a “near-orthogonal” planar drawing in this work?","Question",{"text":112,"@type":113},"A drawing is “close to orthogonal” where closeness is measured by the number of faces that are not orthogonal polygons.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"Why does the paper start from the orthogonal planar drawing limitation?",{"text":117,"@type":113},"Orthogonal planar drawings correspond to planar graphs with maximum degree at most four, since a planar graph has an orthogonal drawing iff its maximum degree is 4.",{"name":119,"@type":110,"acceptedAnswer":120},"What is the computational complexity of testing near-orthogonal drawings with few non-orthogonal faces?",{"text":121,"@type":113},"For triconnected graphs (with unique planar embedding), deciding whether a planar polyline drawing exists with at most h non-orthogonal faces is NP-complete.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},665055,1793287338,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":19,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":46,"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":129,"read_time":31},962093912154,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","arXiv :2607 .21305v1 [ cs .CG] 23 Jul 2026  \nBeyond Degree Four:  \nNear-Orthogonal Planar Drawings  \nPatrizio Angelini \\#   \nJohn Cabot University, Rome, Italy Sabine Cornelsen \\#  University of Konstanz, Germany Giordano Da Lozzo \\#  Roma Tre University, Italy Seok-Hee Hong \\#  University of Sydney, Australia  \nIgnaz Rutter \\#   \nUniversity of Passau, Germany  \n~~ Abstract ~~  \nOrthogonal planar drawings constitute a classical and mainstream research topic in graph drawing due to their clarity and wide applicability. In an orthogonal planar drawing of a graph, each face is represented as an orthogonal polygon, that is, a polygon whose edges are either horizontal or vertical. Yet a planar graph admits such a representation if and only if its maximum degree is at most four.  \nIn this paper, we consider planar polyline drawings of graphs with unrestricted maximum degree. We focus on drawings that are “close to orthogonal”, where closeness is measured by the number of faces that are not orthogonal polygons. We show that, even when the input graph is triconnected and thus has a unique planar embedding, the problem of testing whether there exists a planar polyline drawing with at most h non-orthogonal faces is NP-complete. Motivated by this computational hardness, we study parameterized and approximation algorithms. In the fixed-embedding setting, we prove that the problem admits linear-time FPT algorithms parameterized by (i) the outerplanarity index and (ii) the natural parameter h. In addition, we provide an FPT algorithm parameterized by the treewidth and a polynomial-time approximation scheme. In the variable-embedding setting, we give an FPT algorithm parameterized by treewidth for biconnected graphs.  \nGenerative AI Declaration Generative AI was solely used to improve the writing quality of some paragraphs in the preparation of this article.  \n2012 ACM Subject Classification Theory of computation → Fixed parameter tractability; Theory of computation → Computational geometry; Mathematics of computing → Graph algorithms  \nKeywords and phrases planar orthogonal drawings, unbounded degree, FPT, PTAS, treewidth Category Track 1, Long Paper  \nAcknowledgements This work started at the Bertinoro Workshop on Graph Drawing BWGD 2026 .  \n 1  Introduction  \nOrthogonal planar drawings are among the most classic and extensively studied paradigmsin graph drawing. In this model, each vertex is represented by a point, while each edge is depicted as an orthogonal polyline, i.e., a sequence of horizontal and vertical segments. This representation has proven particularly useful in domains such as circuit design, software engineering, database design, and of course, network visualization [6, 22 , 29 , 37 , 38] . This enduring popularity stems from the clear visual structure created by the combination of  \n2 Beyond Degree Four: Near-Orthogonal Planar Drawings  \nplanarity and axis-aligned edge routes, which make connections much easier to follow and interpret than in more general planar drawing styles.  \nA large body of work has identified the number of bends as a key factor influencing readability and visual quality in orthogonal drawings. Experimental results by Bertolazzi, Di Battista, and Didimo [12] indicate that minimizing bends has a positive impact on several quality measures of drawing effectiveness. The algorithmic study of orthogonal graph drawings has therefore, for decades, focused on bend-minimum and bend-bounded drawings and on the efficient computation of such drawings. In the variable embedding setting, Garg and Tamassia [32] showed that testing whether a graph admits a rectilinear planar drawing, i.e., an orthogonal planar drawing without bends, is NP-complete, confirming a conjecture of Storer [42], and that approximating the minimum number of bends for ann-vertex graph with an O (n1−ϵ ) error is NP-hard for any ϵ > 0. At the same time, several positive results are known for important restricted graph classes, such as planar gra","cbCaigINbzLVWK4E","https://ap.wps.com/l/cbCaigINbzLVWK4E","pdf",1011238,"English","# Introduction\n# Beyond Degree Four: Near-Orthogonal Planar Drawings","[{\"question\":\"What defines a “near-orthogonal” planar drawing in this work?\",\"answer\":\"A drawing is “close to orthogonal” where closeness is measured by the number of faces that are not orthogonal polygons.\"},{\"question\":\"Why does the paper start from the orthogonal planar drawing limitation?\",\"answer\":\"Orthogonal planar drawings correspond to planar graphs with maximum degree at most four, since a planar graph has an orthogonal drawing iff its maximum degree is 4.\"},{\"question\":\"What is the computational complexity of testing near-orthogonal drawings with few non-orthogonal faces?\",\"answer\":\"For triconnected graphs (with unique planar embedding), deciding whether a planar polyline drawing exists with at most h non-orthogonal faces is NP-complete.\"}]","Beyond Degree Four - Near-Orthogonal Planar Drawings | PDF"]