[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-649825-105":59,"doc-detail-649825-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","extending-orthogonal-planar-graph-drawings-is-fixed-parameter-tractable","Extending Orthogonal Planar Graph Drawings is Fixed-Parameter Tractable","","Extends partial drawings of graphs by determining whether a bend-minimal orthogonal drawing of a planar connected graph can be completed into a full drawing while preserving orthogonal representation constraints. Although the extension problem is known to be NP-hard, the work proves fixed-parameter tractability when parameterized by the size of the missing subgraph. The approach combines geometric and combinatorial techniques, including a bend-equivalent region representation for vertex placement, a treewidth bound for an auxiliary graph, and a global point-grid to discretize bend and vertex locations into bounded subgrids.",{"@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/extending-orthogonal-planar-graph-drawings-is-fixed-parameter-tractable/649825/",{"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/extending-orthogonal-planar-graph-drawings-is-fixed-parameter-tractable/649825.png","ImageObject",300,407,{"name":92,"@type":93},"EvelynWrites","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-25","2026-10-24",true,{"@type":102,"interactionType":103,"userInteractionCount":8},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What extension problem is studied in the paper?","Question",{"text":112,"@type":113},"The paper studies extending a bend-minimal orthogonal drawing of a planar connected graph from a partial drawing to a full orthogonal drawing while respecting orthogonal representation constraints.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"Why is the problem challenging?",{"text":117,"@type":113},"The orthogonal bend-minimal extension problem is NP-hard in general, and the paper situates it within known NP-complete hardness when the pre-drawn part is empty.",{"name":119,"@type":110,"acceptedAnswer":120},"What is the main algorithmic contribution?",{"text":121,"@type":113},"It establishes fixed-parameter tractability with the parameter being the size of the missing subgraph, using new geometric-combinatorial ingredients such as a bend-equivalent region representation, a treewidth bound, and a point-grid discretization strategy.","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},649825,1792899751,{"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":8,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145},687212989558,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","arXiv :2302 . 10046v1 [ cs .CG] 20 Feb 2023  \nExtending Orthogonal Planar Graph Drawings is Fixed-Parameter Tractable  \nSujoy Bhore !   \nIndian Institute of Technology Bombay, India Robert Ganian !   \nTechnische Universität Wien, Austria  \nLiana Khazaliya !  \nTechnische Universität Wien, Austria Fabrizio Montecchiani !  Università degli Studi di Perugia, Italy Martin Nöllenburg !  Technische Universität Wien, Austria  \n~~ Abstract ~~  \nThe task of finding an extension to a given partial drawing of a graph while adhering to constraints on the representation has been extensively studied in the literature, with well-known results providing efficient algorithms for fundamental representations such as planar and beyond-planar topological drawings. In this paper, we consider the extension problem for bend-minimal orthogonal drawings of planar connected graphs, which is among the most fundamental geometric graph drawing representations. While the problem was known to be NP-hard, it is natural to consider the case where only a small part of the graph is still to be drawn. Here, we establish the fixed-parameter tractability of the problem when parameterized by the size of the missing subgraph. Our algorithm is based on multiple novel ingredients which intertwine geometric and combinatorial arguments. These include the identification of a new graph representation of bend-equivalent regions for vertex placement in the plane, establishing a bound on the treewidth of this auxiliary graph, and a global point-grid that allows us to discretize the possible placement of bends and vertices into locally bounded subgrids for each of the above regions.  \n2012 ACM Subject Classification Theory of computation → Computational geometry; Mathematics of computing → Graph algorithms  \nKeywords and phrases orthogonal drawings, bend minimization, extension problems, parameterized complexity  \n 1  Introduction  \nExtending partial drawings of graphs while preserving certain desirable properties such as planarity is an algorithmic problem that received considerable attention in the last decade in graph theory, graph drawing, and computational geometry. Drawing extension problems are motivated, for instance, by visualizing networks, in which certain subgraphs represent important motifs that require a specific drawing, or by visualizing dynamic networks, in which new edges and vertices must be integrated in an existing, stable drawing. Generally speaking, we are given a graph G and a (typically connected) subgraph H of G with a drawing Γ(H), which is called a partial drawing of G. The drawing Γ(H) typically satisfies certain topological or geometric properties, e.g., planarity, upward planarity, or 1-planarity, and the goal of the corresponding extension problem is to extend Γ(H) to a drawing Γ(G) of the whole graph G (if possible) by inserting the missing vertices and edges into Γ(H) while maintaining the required drawing properties.  \nA fundamental result in this line of research is the work of Angelini et al. [1], who showed  \n2 Extending Orthogonal Planar Graph Drawings is FPT  \nthat for planar graphs with a given partial planar drawing, the extension problem can be solved in linear time, thus matching the time complexity of unconstrained planarity testing. In fact, there is also a corresponding combinatorial characterization of planar graphs with extensible partial planar drawings via forbidden substructures [25] . In contrast to the above results, which consider topological graph embeddings, the planar drawing extension problem is NP-hard in its geometric variant, where one has to decide if a partial planar straight-line drawing Γ(H) can be extended to a planar straight-line drawing of G [31] .  \nIn this paper, we study the geometric drawing extension problem arising in the context of one of the most fundamental graph drawing styles: orthogonal drawings [12,16,19,30] . Ina planar orthogonal drawing, edges are represented as polylines comprised of (one or more) ho","cbCaio5IvZsRjfJR","https://ap.wps.com/l/cbCaio5IvZsRjfJR","pdf",1154253,28,"English","# Introduction\n## Background on extension problems\n## Orthogonal drawings and bend-minimal objectives\n## Complexity results and related work","[{\"question\":\"What extension problem is studied in the paper?\",\"answer\":\"The paper studies extending a bend-minimal orthogonal drawing of a planar connected graph from a partial drawing to a full orthogonal drawing while respecting orthogonal representation constraints.\"},{\"question\":\"Why is the problem challenging?\",\"answer\":\"The orthogonal bend-minimal extension problem is NP-hard in general, and the paper situates it within known NP-complete hardness when the pre-drawn part is empty.\"},{\"question\":\"What is the main algorithmic contribution?\",\"answer\":\"It establishes fixed-parameter tractability with the parameter being the size of the missing subgraph, using new geometric-combinatorial ingredients such as a bend-equivalent region representation, a treewidth bound, and a point-grid discretization strategy.\"}]","Extending Orthogonal Planar Graph Drawings is Fixed-Parameter Tractable | PDF",1792808817,71]