[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84162-en":3,"doc-seo-84162-105":29,"detail-sidebar-cat-0-en-105":91},{"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84162,2336464648746,"Skyler","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Layer-Respecting Linear Graph Layouts","Layer-Respecting Linear Graph Layouts presents algorithms for graph drawing that minimize edge crossings while keeping vertices and layers consistent with an imposed ordering. It studies three layout variants: layer-respecting arc diagrams and layer-respecting linear cylindric drawings, defined so layers appear in order on a single line and vertices are grouped by their layers. Although the general arc-diagram crossing problem is NP-hard, fixed-parameter tractable linear-time methods are given using the layered graph’s width, derived from BFS layering bounds.","Layer-Respecting Linear Graph Layouts  \nAlvin Chiu∗ David Eppstein† Michael T. Goodrich‡ Songyu (Alfred) Liu§  \narXiv :2607 .06968v 1 [ cs .DS] 8 Jul 2026  \nAbstract  \nWe show how to visualize a graph, G = (V, E), as a layered drawing, layer-respecting arc diagram, or layerrespecting linear cylindric drawing with a minimum number of edge crossings, where layer-respecting means that layers appear in order on a single line and vertices are grouped by their layers. Even though this problem is NP-hard for general arc diagrams, we show how to create such diagrams with fixed-parameter tractable linear-time algorithms, where the parameter that allows this is the width of a layered graph. Such a layered graph can be obtained from a breadth-first search (BFS), in which case the width is upper bounded by a graph width parameter called the BFS width.  \n1 Introduction  \nA layering of a graph G = (V, E) is a partition,{L1 , L2 ,..., LH }, of V into disjoint subsets, which are referred to as layers, such that S Li = V and for every edge (v, w) ∈ E, if v ∈ Li and w ∈ Lj , then |i − j| ≤ 1; see, e.g., [14, 5] . The width of a layered graph is the number of vertices in its largest layer [18] . One nice property of layerings of a connected graph is that removing all the vertices in any layer that is not the topmost or bottommost layer will disconnect the graph. The pioneering work on graph separators by Lipton and Tarjan [21] used a layering of a connected graph G computed from a breadth-first search (BFS) on an arbitrarily chosen vertex in G by defining each layer by its distance from the starting vertex. We define such a layering formed using the BFS tree from some vertex as a BFS layering. Note that a graph layering is not necessarily a BFS layering, but every BFS layering is a graph layering. To see the latter, suppose we have a BFS layering with starting vertex r. V is naturally partitioned into disjoint subsets by distances from r. For any edge (v, w) ∈ E where v ∈ Li and w ∈ Lj , we have |i−j| ≤ 1 by the reverse triangle inequality for the distances between v , w, and r. See Figure 1 .  \nThe work on layered drawings of graphs by Sugiyama, Tagawa, and Toda [25] pioneered the graph drawing paradigm now known as a “Sugiyama-style” drawing.  \n∗ University of California, Irvine, [chiua13@uci.edu](chiua13@uci.edu)[ ](chiua13@uci.edu)†University of California, Irvine, [eppstein@uci.edu](eppstein@uci.edu)[ ](eppstein@uci.edu)‡University of California, Irvine, [goodrich@uci.edu](goodrich@uci.edu)  \n§ University of California, Irvine, [songyul4@uci.edu](songyul4@uci.edu)  \nFigure 1: Left: a layered drawing of a layered graph. The partition indicated by Li is a valid layering but not a BFS layering. Right: a Sugiyama-style drawing of a directed graph. The edge between the topmost vertex and bottommost vertex spans multiple layers. Thus, the partition indicated by Li is not a valid layering for the graph on the right.  \nThis style is for directed graphs, and in this style, vertices are drawn as points on horizontal lines and each line is one layer; all edges point downwards; straightline segments are used for edges joining two points in consecutive layers and polygonal chains may be used for edges joining two points spanning multiple layers. See, e.g., [17] . Strictly speaking, a Sugiyama-style drawing does not allow for edges between vertices in the same layer, but we do not make this restriction in this paper. See Figure 1 . In a Sugiyama-style drawing, the number of layers is lower bounded by the number of vertices on a longest path, since each vertex on this path must be on a different layer [23] . With any BFS layering, the number of layers is upper bounded by the number of vertices on a longest path. Thus, we are interested inan alternate graph drawing paradigm. Furthermore, we also study linear graph layouts [7, 2], in particular arc diagrams and linear cylindric drawings. As it turns out, under certain constraints, crossing minimization for ","cbCaigS5gjjuaVkg","https://ap.wps.com/l/cbCaigS5gjjuaVkg","pdf",523833,1,10,"English","en",105,"# Introduction\n# Layered drawings\n## Sugiyama-style drawings\n# Arc diagrams","[{\"question\":\"What does it mean for a drawing to be “layer-respecting” in this work?\",\"answer\":\"Layers must appear in order on a single horizontal line, and vertices are grouped so each vertex layer forms consecutive blocks in the linear ordering. Edges then follow the defined layout style without violating the layer order constraint.\"},{\"question\":\"Is the minimum edge-crossing problem easy or hard in general?\",\"answer\":\"For general arc diagrams, minimizing crossings is NP-hard. The work shows tractable solutions under the added constraint of layer-respecting layouts.\"},{\"question\":\"What parameter enables the fixed-parameter tractable linear-time algorithms?\",\"answer\":\"The key parameter is the width of a layered graph, defined as the number of vertices in the largest layer. When the layering comes from BFS, the width is bounded by a BFS-width graph parameter.\"}]",1784193546,25,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"layer-respecting-linear-graph-layouts","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/layer-respecting-linear-graph-layouts/84162/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What does it mean for a drawing to be “layer-respecting” in this work?","Question",{"text":75,"@type":76},"Layers must appear in order on a single horizontal line, and vertices are grouped so each vertex layer forms consecutive blocks in the linear ordering. Edges then follow the defined layout style without violating the layer order constraint.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Is the minimum edge-crossing problem easy or hard in general?",{"text":80,"@type":76},"For general arc diagrams, minimizing crossings is NP-hard. The work shows tractable solutions under the added constraint of layer-respecting layouts.",{"name":82,"@type":73,"acceptedAnswer":83},"What parameter enables the fixed-parameter tractable linear-time algorithms?",{"text":84,"@type":76},"The key parameter is the width of a layered graph, defined as the number of vertices in the largest layer. 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