[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-665057-en":3,"doc-seo-665057-105":31,"detail-sidebar-cat-0-en-105":97},{"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},665057,962094243231,"RizkyPratama_92","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Smooth Orthogonal Drawings of Planar Graphs - Paper","Smooth orthogonal layouts of planar graphs require each edge to alternate axis-aligned segments and circular arcs while maintaining common axis-aligned tangents. This work investigates smooth orthogonal drawings with low edge complexity, defining smooth complexity k (SCk) as the existence of a smooth orthogonal drawing where every edge uses at most k segments. The results establish SC2-layouts for all 4-planar graphs and show cubic area suffices for 3-planar graphs, with both positive and negative findings for 4-outerplane graphs.","Smooth Orthogonal Drawings of Planar Graphs?  \nMd. Jawaherul Alam 1 , Michael A. Bekos2 , Michael Kaufmann2 , Philipp Kindermann3 , Stephen G. Kobourov 1 , and Alexander Wolff3  \n1 Department of Computer Science, University of Arizona, USA  \nfmjalam,[kobourov](kobourovg@cs.arizona.edu)[g](kobourovg@cs.arizona.edu)[@cs.arizona.edu](kobourovg@cs.arizona.edu)[ ](kobourovg@cs.arizona.edu)2 Wilhelm-Schickhard-Institut f¨ur Informatik, Universitt T¨ubingen, Germany fbekos,[mk](mkg@informatik.uni-tuebingen.de)[g](mkg@informatik.uni-tuebingen.de)[@informatik.uni-tuebingen.de](mkg@informatik.uni-tuebingen.de)[ ](mkg@informatik.uni-tuebingen.de)3 Lehrstuhl f¨ur Informatik I, Universitt W¨urzburg, Germany  \n[http://www1.informatik.uni-wuerzburg.de/en/staff](http://www1.informatik.uni-wuerzburg.de/en/staff)  \nAbstract. In smooth orthogonal layouts of planar graphs, every edge is an alternating sequence of axis-aligned segments and circular arcs with common axisaligned tangents. In this paper, we study the problem of finding smooth orthogonal layouts of low edge complexity, that is, with few segments per edge. We say that a graph has smooth complexity k—for short, an SCk-layout—if it admits a smooth orthogonal drawing of edge complexity at most k.  \nOur main result is that every 4-planar graph has an SC2-layout. While our drawings may have super-polynomial area, we show that for 3-planar graphs, cubic area suffices. We also show that any biconnected 4-outerplane graph has an SC1-layout. On the negative side, we demonstrate an infinite family of biconnected 4-planar graphs that require exponential area for an SC1-layout. Finally, we present an infinite family ofbiconnected 4-planar graphs that do not admit an SC1-layout.  \n1 Introduction  \nIn the visualization of technical networks such as the structure of VLSI chips [8] or UML diagrams [10] there is a strong tendency to draw edges as rectilinear paths. The problem of laying out networks in such a way is called orthogonal graph drawing and has been studied extensively. For drawings of (planar) graphs to be readable, special care is needed to keep the number of bends small. In a seminal work, Tamassia [11] showed that one can efficiently minimize the total number of bends in orthogonal layouts of embedded 4-planar graphs, that is, planar graphs of maximum degree 4 whose combinatorial embedding (the cyclic order of the edges around each vertex) is given. In contrast to this, minimizing the number of bends over all embeddings of a 4-planar graph is NP-hard [6] .  \n? Research of M. J. Alam and S. G. Kobourov is supported in part by NSF grants CCF-1115971 and DEB 1053573 . The work of M. A. Bekos is implemented within the framework of the Action “Supporting Postdoctoral Researchers” of the Operational Program “Education and Lifelong Learning”(Action's Beneficiary: General Secretariat for Research and Technology), and is co-financed by the European Social Fund (ESF) and the Greek State. M. Kaufmann as well as Ph. Kindermann and A. Wolff acknowledge support by the ESF EuroGIGA project GraDR (DFG grants Ka 812/16-1 and Wo 758/5-1, respectively) .  \n2 Md. Alam et al.  \nIn a so far unrelated line of research, circular-arc drawings of graphs have become a popular matter of research in the last few years. Inspired by American artist Mark Lombardi (1951–2000), Duncan et al. [4] introduced and studied Lombardi drawings, which are circular-arc drawings with the additional requirement of perfect angular resolution, that is, for each vertex, all pairs of consecutive edges form the same angle. Among others, Duncan et al. treat drawings of d-regular graphs where all vertices have to lie on one circle. They show that under this restriction, for some subclasses, Lombardi drawings can be constructed efficiently, whereas for the others, the problem is NP-hard. They also show [5] that ordered trees can always be Lombardi drawn in polynomial area, whereas straight-line drawings with perfect resolution may need exponential ","cbCaiiizvKUBPwRi","https://ap.wps.com/l/cbCaiiizvKUBPwRi","pdf",728456,4,1,12,"English","en",105,"# Abstract\n# Introduction\n# Smooth Orthogonal Layout Model\n## Complexity Definitions\n# Related Work\n## Orthogonal and Lombardi Drawings\n# Results and Contributions","[{\"question\":\"What is an SCk-layout in this paper?\",\"answer\":\"A graph has smooth complexity k (an SCk-layout) if it admits a smooth orthogonal drawing where every edge has edge complexity at most k.\"},{\"question\":\"What is the main result about 4-planar graphs?\",\"answer\":\"Every 4-planar graph has an SC2-layout.\"},{\"question\":\"What kinds of outcomes does the paper provide for 4-outerplane graphs regarding SC1-layouts?\",\"answer\":\"It shows any biconnected 4-outerplane graph has an SC1-layout, but it also proves there are infinite families of biconnected 4-planar graphs that require exponential area for an SC1-layout and others that do not admit an SC1-layout.\"}]","Smooth Orthogonal Drawings of Planar Graphs - 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