[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83193-en":3,"doc-seo-83193-105":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},83193,1374391974468,"Eden","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Geodetic sets for directed acyclic planar geodetic graphs","A set of vertices S in a directed graph is geodetic if every vertex lies on a shortest path from some vertex of S to another vertex of S. A directed graph is geodetic if every ordered pair admits at most one shortest path. The decision problem—given a directed acyclic planar geodetic graph G and integer k, whether G has a geodetic set with at most k vertices—is proved NP-complete. Consequently, strong and monitoring variants are also NP-complete. For directed acyclic series-parallel graphs, minimum geodetic and edge-geodetic set sizes are computable in linear time.","arXiv :2607 .07107v1 [math .CO] 8 Jul 2026  \nGeodetic sets for directed acyclic planar geodetic graphs  \nBenedikt Georg Hein 1 and Egon Wanke2  \n1 ,2 Heinrich-Heine-Universit¨at D¨usseldorf, Germany  \n[1](1 BeHei111@hhu.de)[ BeHei111@hhu.de](1 BeHei111@hhu.de) , [2](2 Egon.Wanke@hhu.de)[ Egon.Wanke@hhu.de](2 Egon.Wanke@hhu.de)  \nAbstract  \nA set of vertices S of a directed graph G is geodetic if every vertex of G lies on a shortest path from a vertex of S to a vertex of S. A directed graph is geodetic if there is at most one shortest path from every vertex of G to every vertex of G. We prove the NP-completeness of the following decision problem. Given a directed acyclic planar geodetic graph G and an integer k, does G have a geodetic set with at most k vertices? This implies that the question of whether G has a strong or a monitoring geodetic set with at most k vertices is also NP-complete for directed acyclic planar geodetic graphs. Furthermore, we prove that the number of vertices in a minimum geodetic set and the number of vertices in a minimum edge geodetic set can be computed in linear time for directed acyclic series-parallel graphs.  \nKeywords : Geodetic graph, geodetic number, strong geodetic set, monitoring geodetic set  \n1 Introduction  \nWe define various geodetic sets for directed graphs that are determined in a similar way. To avoid repetitions as much as possible, the edge variants are defined using the expressions given in parentheses.  \nFor a graph G = (V, E), let V (G) = V be the set of vertices and E (G) = E ⊆ V × V be the set of edges of G. The length of a path p is the number of its edges. Sometimes we represent paths as a sequence of vertices, sometimes as a sequence of edges, depending on what information about the paths is relevant to our considerations. A vertex w (an edge e) is covered by a path p if p contains the vertex w (the edge e) . A vertex w (an edge e) is covered by a set of vertices S if there exist two vertices u, v ∈ S and a shortest path from u tov that covers the vertex w (the edge e) .  \nA vertex set S ⊆ V (G) is a geodetic set (an edge geodetic set) of G if every vertex w ∈ V (G) (every edge e ∈ E (G)) is covered by a shortest path from a vertex of S to a vertex of S. The vertices u ∈ S are always covered by the vertices in S, because the path consisting of the single vertex u is always a shortest path from u to u.  \nA vertex set S ⊆ V (G) is called a strong geodetic set (strong edge geodetic set) of G if there exists a set of shortest paths P, each from some vertex u ∈ S to some vertex v ∈ S such that every vertex in V (G) (every edge in E (G)) is covered by at least one of these paths. It should be noted here that, for every pair of nodes u, v ∈ S, the set P may contain at most one path from u to v.  \nA vertex set S ⊆ V (G) is a monitoring geodetic set (monitoring edge geodetic set) of G if for each vertex w ∈ V (G) (edge e ∈ E (G)) there is a pair of vertices u, v ∈ S such that vertex w (edge e) is covered by all shortest path from u to v.  \nLet GS(G), SGS(G), MoGS(G), EGS(G), SEGS(G) and MoEGS(G) be the set of all geodetic sets, strong geodetic sets, monitoring geodetic sets, edge geodetic sets, strong edge geodetic sets and monitoring edge geodetic sets, respectively, of a graph G. If every vertex has at least one outgoing or at least one incoming edge then every edge geodetic set is a geodetic set, every strong edge geodetic set is a strong geodetic set and every monitoring edge geodetic set is a monitoring geodetic set. That is, for every graph G satisfying this condition it holds that EGS(G) ⊆ GS(G), SEGS(G) ⊆ SGS(G) and MoEGS(G) ⊆ MoGS(G) . Furthermore, every strong geodetic (strong edge geodetic) set obviously is a geodetic set (an edge geodetic set) and every monitoring geodetic set (monitoring edge geodetic set) is a strong geodetic set (strong edge geodetic set, respectively) . The second statement follows from the following fact: If a vertex w (edge e) lies on all shortest ","cbCairaghh2fLtjN","https://ap.wps.com/l/cbCairaghh2fLtjN","pdf",617839,4,1,24,"English","en",105,"# Abstract\n# Introduction\n# Preliminaries\n# (Further sections) Geodetic sets and complexity results","[{\"question\":\"What does it mean for a vertex set S to be geodetic in a directed graph?\",\"answer\":\"S is geodetic if every vertex of the graph lies on a shortest path from some vertex in S to some other vertex in S.\"},{\"question\":\"How is a directed graph defined as geodetic in this work?\",\"answer\":\"A directed graph is geodetic if, for every pair of vertices, there is at most one shortest path from the first vertex to the second.\"},{\"question\":\"What complexity result is proved for directed acyclic planar geodetic graphs?\",\"answer\":\"The decision problem of whether such a graph has a geodetic set of size at most k is NP-complete, and the strong and monitoring variants with the same bound are also NP-complete.\"}]",1784185869,60,{"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},"geodetic-sets-for-directed-acyclic-planar-geodetic-graphs","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/geodetic-sets-for-directed-acyclic-planar-geodetic-graphs/83193/",{"url":52,"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","2026-07-24","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 vertex set S to be geodetic in a directed graph?","Question",{"text":75,"@type":76},"S is geodetic if every vertex of the graph lies on a shortest path from some vertex in S to some other vertex in S.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is a directed graph defined as geodetic in this work?",{"text":80,"@type":76},"A directed graph is geodetic if, for every pair of vertices, there is at most one shortest path from the first vertex to the second.",{"name":82,"@type":73,"acceptedAnswer":83},"What complexity result is proved for directed acyclic planar geodetic graphs?",{"text":84,"@type":76},"The decision problem of whether such a graph has a geodetic set of size at most k is NP-complete, and the strong and monitoring variants with the same bound are also NP-complete.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,109,114,119,122,127,130,134],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"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":29,"slug":108},5,"Comic","comic",{"id":110,"doc_module":4,"doc_module_name":46,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]