[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83056-en":3,"doc-seo-83056-105":30,"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":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":13,"seo_description":14,"update_tm":28,"read_time":29},83056,13056703019404,"Miles","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Breadth-First Search in Succinct Planar Graphs","An encoding for planar graphs is defined as succinct when it represents n-vertex graphs using Z_G(n)+o(Z_G(n)) bits, and compact when using O(Z_G(n)) bits, where Z_G(n) is the information-theoretic minimum and for planar graphs Z_G(n)=Θ(n). The work presents a succinct planar encoding that enables running breadth-first search directly on the representation while keeping the BFS tree for constant-time queries. It also supports related plane/dual traversal and enables space-efficient planar separator computations and further applications.","Breadth-First Search in Succinct Planar Graphs  \nJohannes Meintrup *  \narXiv :2607 .0622 1v 1 [ cs .DS] 7 Jul 2026  \nAbstract  \nAn encoding for a graph class G is called succinct if it uses ZG (n)+o(ZG (n)) bits, where ZG (n) is the information-theoretic minimum number of bits needed to distinguish between n-vertex graphs of G ; it is called compact if it uses O (ZG (n)) bits. For planar graphs, ZG (n) = Θ(n) . We present a succinct encoding of planar graphs that supports executing a breadth-first search directly on the encoding. The succinct encoding can be constructed in expected O (n) time using O (n) bits during construction; a compact variant can be constructed in deterministic O (n) time using O (n) bits. Once the encoding is constructed, a BFS from any start vertex can be computed in O (n) time using o(n) additional bits, including the space needed to represent the BFS tree. The resulting BFS tree T remains available for standard tree operations, such as traversal, parent and child queries, layer queries, and lowest common ancestor queries, in constant time per query or output element. The encoding also supports standard graph queries. For plane graphs G = (V, E), we provide traversal of the interdigitating tree ˆT, i.e., the spanning tree of the dual graph whose edges correspond to E \\E(T ) .  \nAs our main application, we implement the well-known planar separator theorem in a space-efficient way. For biconnected plane graphs, our encoding allows us to compute a balanced separator of size O( √n)  in O (n) time using o(n) additional bits. Along the way, we show that biconnected plane graphs encoded by our representation can be triangulated in expected O (n) time and o(n) bits in the succinct variant, or in deterministic O (n) time using O (n) bits in the compact variant. Further applications include computation of a tree decomposition of width O (d) where d is the diameter of the plane graph at hand and testing for bipartiteness. Finally, all results that do not rely on a plane embedding generalize to separable graph classes.  \n* THM University of Applied Sciences Mittelhessen, Giessen, Germany. Email: [johannes.meintrup@mni.thm.de](johannes.meintrup@mni.thm.de. Funded by)[. Funded by](johannes.meintrup@mni.thm.de. Funded by)[ ](johannes.meintrup@mni.thm.de. Funded by)[the Deutsche Forschungsgemeinschaft](the Deutsche Forschungsgemeinschaft) ([DFG](DFG), [German Research Foundation](German Research Foundation))– 571642628.  \n1 Introduction  \n1.1. Breadth-first search (BFS) is, alongside depth-first search (DFS), one of the most fundamental graph traversal algorithms. Throughout the paper, we use n to denote the number of vertices of a graph and m to denote the number of edges. A standard BFS running in O(n +m) time stores a queue of vertices and a visitation status for every vertex, and therefore uses Θ (nlog n) bits in the word-RAM model.  \n1.2. Several works study space-efficient BFS and DFS algorithms that use (almost) linear space in bits while retaining (almost) linear running time [3, 5, 12, 17] . Related work on reachability pushes the space below linear, at the cost of large polynomial running time [31, 22] . The space-efficient BFS and DFS algorithms above primarily output the vertices in visitation order; they do not keep the resulting BFS or DFS tree available as a data structure.  \n1.3. This missing structural access is a serious limitation when a traversal is used as a subroutine in more complex applications. Standard graph algorithms often rely not only on the order in which vertices are visited, but also on direct access to the resulting tree: parent and child queries, depths or layers, ancestor queries, and more. In the space-efficient setting, such information cannot simply be stored for all vertices. Consequently, space-efficient algorithms using BFS or DFS as subroutines often require substantial modifications to workaround this lack of queries. For example, space-efficient algorithms for biconnected c","cbCaimav2W6utFEL","https://ap.wps.com/l/cbCaimav2W6utFEL","pdf",318283,2,1,22,"English","en",105,"# Abstract\n# Introduction\n## Breadth-first search (BFS)\n## Space-efficient BFS/DFS and structural access\n## Prior work on DFS in succinct encodings\n## Main contribution: succinct BFS with accessible BFS tree\n## Technical ideas and complexity overview","[{\"question\":\"What does “succinct” mean for graph encodings in this work?\",\"answer\":\"A graph class encoding is succinct if it uses Z_G(n)+o(Z_G(n)) bits, where Z_G(n) is the minimum number of bits needed to distinguish n-vertex graphs in that class. For planar graphs, the information-theoretic minimum satisfies Z_G(n)=Θ(n).\"},{\"question\":\"How does the proposed encoding support breadth-first search?\",\"answer\":\"It supports executing BFS directly on the succinct planar graph encoding from any start vertex. After constructing the encoding, a BFS can be computed in O(n) time using o(n) extra bits while keeping the BFS tree available for standard operations.\"},{\"question\":\"What additional algorithms or applications are enabled beyond BFS?\",\"answer\":\"The work applies the encoding to compute planar separators space-efficiently, including balanced separators of size O(√n) for biconnected plane graphs. It also enables operations such as triangulation, tree decomposition of bounded width tied to diameter, bipartiteness testing, and generalizations to separable graph classes.\"}]",1784184907,55,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"breadth-first-search-in-succinct-planar-graphs","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/breadth-first-search-in-succinct-planar-graphs/83056/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-21","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 “succinct” mean for graph encodings in this work?","Question",{"text":75,"@type":76},"A graph class encoding is succinct if it uses Z_G(n)+o(Z_G(n)) bits, where Z_G(n) is the minimum number of bits needed to distinguish n-vertex graphs in that class. For planar graphs, the information-theoretic minimum satisfies Z_G(n)=Θ(n).","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed encoding support breadth-first search?",{"text":80,"@type":76},"It supports executing BFS directly on the succinct planar graph encoding from any start vertex. After constructing the encoding, a BFS can be computed in O(n) time using o(n) extra bits while keeping the BFS tree available for standard operations.",{"name":82,"@type":73,"acceptedAnswer":83},"What additional algorithms or applications are enabled beyond BFS?",{"text":84,"@type":76},"The work applies the encoding to compute planar separators space-efficiently, including balanced separators of size O(√n) for biconnected plane graphs. It also enables operations such as triangulation, tree decomposition of bounded width tied to diameter, bipartiteness testing, and generalizations to separable graph classes.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]