[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82032-en":3,"doc-seo-82032-105":31,"detail-sidebar-cat-0-en-105":93},{"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},82032,7971461740886,"Theodore","https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2",8,"Research & Report","The Parameterised Complexity of Temporal Motif Counting, and a Lovász-Style Isomorphism Theorem","Studies the structural expressivity and parameterised complexity of counting homomorphisms from small temporal patterns to large temporal graphs. A temporal pattern is a graph equipped with a partial order on its edges, and homomorphisms must preserve edges while respecting the temporal constraints. Results include a Lovász-style isomorphism theorem, an FPT approach via the toadwidth measure, and a parameterised complexity dichotomy for totally ordered temporal patterns.","The Parameterised Complexity of Temporal Motif Counting, and a Lovsz-Style Isomorphism Theorem  \narXiv :2607 .086 14v 1 [ cs .CC] 9 Jul 2026  \nJayakrishnan Madathil \\# Indian Institute of Technology Palakkad  \nKitty Meeks \\#  \nSchool of Computing Science, University of Glasgow  \nMarc Roth \\#  \nSchool of Electronic Engineering and Computer Science, Queen Mary University of London  \n~~ Abstract ~~  \nWe study the structural expressivity and the parameterised complexity of counting homomorphisms from small temporal patterns to large temporal graphs. Here, a temporal pattern 􀀥 consists of a graph together with a partial order on its edges, and a homomorphism from 􀀥 to a temporal graph must not only preserve edges, but also satisfy the temporal constraints imposed by the partial order of the edge set of the pattern.  \nThe main results of this work are three-fold:  \n(1) We prove a temporal Lovsz-style theorem, stating that two temporal graphs are isomorphic (under a natural definition of temporal isomorphisms) if and only if they have the same number of homomorphisms from all temporal patterns.  \n(2) We introduce a cliquewidth-based measure on temporal patterns, called the temporally order-augmented dual width, the toadwidth for short, and show that counting temporal homomorphisms is fixed-parameter tractable for temporal patterns of bounded toadwidth.  \n(3) We provide a parameterised complexity dichotomy with an explicit tractability criterion for counting homomorphisms from totally ordered temporal patterns, classified along their underlying graph structure.  \nThe methods and tools invoked for proving (1)-(3) vary significantly: The proof of the Lovsz-style Theorem is obtained by combining Lovsz’ original argument with an inclusion-exclusion construction to deal with temporal equalities and inequalities. The FPT algorithm in (2) is obtained by an involved dynamic programming algorithm along toadwidth. Finally, the upper bound for the dichotomy in (3) relies on the toadwidth-based algorithm in (2), and the lower bound follows from a reduction from the clique problem by embedding the temporal pattern into a grid, the connections of which are either formed by edges or by temporal constraints.  \nJ. Madathil, K. Meeks, and M. Roth 1  \n 1  Introduction  \nTemporal graphs model networks the connections of which are only available at certain points, or during certain intervals, of time. They find applications in the analysis of protein-protein interaction networks [23], social networks [41], and in machine learning [31, 40], only to name a few examples; we refer the reader to the survey of Holme and Saramki [26] for a comprehensive exposition.  \nFollowing the success of “Network Motifs” in static (non-temporal) graphs [34] recent years have seen a flurry of applied results on temporal motif counting problems [36, 30, 39, 29]: in a nutshell, given a pattern  \n􀀥 and a temporal graph Γ, the task is to count the number of occurrences of 􀀥 in Γ . Similarly as in the static case, it has been observed that frequencies of temporal patterns correlate with global features of the temporal network. Despite their relevance in practical applications witnessed by the previous works, we find a notable gap when it comes to our theoretical understanding on the inherent complexity of temporal motif counting problems: while there are results for selected patterns, such as temporal walks [17], and temporal stars [29], the state of the art is far away from a comprehensive understanding of the complexity of counting temporal patterns in general. This stands in sharp contrast to the static, non-temporal, case, for which we know deep dichotomy results w.r.t. parameterised and fine-grained complexity theory that determine almost precisely the best possible running times for arbitrary motif counting problems under standard lower bound assumptions [13, 33, 11, 10, 20, 15] .  \nIn this work, we address this gap and present the first comprehensive complexity analysis of ","cbCaimkgQZ3go33Q","https://ap.wps.com/l/cbCaimkgQZ3go33Q","pdf",636174,7,1,54,"English","en",105,"# Introduction\n# The Model: Temporal Graphs and Parameterised Complexity","[{\"question\":\"What problem does the paper address in temporal graphs?\",\"answer\":\"The paper analyzes the complexity of counting homomorphisms from small temporal patterns to large temporal graphs, focusing on how temporal constraints affect counting difficulty.\"},{\"question\":\"How is a temporal pattern defined and what makes a homomorphism valid?\",\"answer\":\"A temporal pattern consists of a graph plus a partial order on its edges. A homomorphism must preserve edges and also satisfy the temporal constraints imposed by that partial order.\"},{\"question\":\"What are the main theoretical contributions of the work?\",\"answer\":\"It proves a Lovász-style theorem for temporal isomorphism, introduces the toadwidth measure to obtain fixed-parameter tractability for bounded toadwidth patterns, and establishes a parameterised complexity dichotomy with a tractability criterion for totally ordered temporal patterns.\"}]","The Parameterised Complexity of Temporal Motif Counting, and a Lovász-Style Isomorphism Theorem | PDF",1784177703,136,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":88,"head_meta":90,"extra_data":92,"updated_unix":29},"the-parameterised-complexity-of-temporal-motif-counting-and-a-lovasz-style-isomorphism-theorem","",{"@graph":37,"@context":87},[38,55,70],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,52],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":51},"https://docshare.wps.com/document/research-report/",3,{"item":53,"name":13,"@type":44,"position":54},"https://docshare.wps.com/document/the-parameterised-complexity-of-temporal-motif-counting-and-a-lovasz-style-isomorphism-theorem/82032/",4,{"url":53,"name":13,"@type":56,"author":57,"headline":13,"publisher":59,"fileFormat":62,"inLanguage":24,"description":14,"dateModified":63,"datePublished":64,"encodingFormat":62,"isAccessibleForFree":65,"interactionStatistic":66},"DigitalDocument",{"name":9,"@type":58},"Person",{"url":42,"name":60,"@type":61},"DocShare","Organization","application/pdf","2026-07-29","2026-07-16",true,{"@type":67,"interactionType":68,"userInteractionCount":20},"InteractionCounter",{"@type":69},"ViewAction",{"@type":71,"mainEntity":72},"FAQPage",[73,79,83],{"name":74,"@type":75,"acceptedAnswer":76},"What problem does the paper address in temporal graphs?","Question",{"text":77,"@type":78},"The paper analyzes the complexity of counting homomorphisms from small temporal patterns to large temporal graphs, focusing on how temporal constraints affect counting difficulty.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"How is a temporal pattern defined and what makes a homomorphism valid?",{"text":82,"@type":78},"A temporal pattern consists of a graph plus a partial order on its edges. A homomorphism must preserve edges and also satisfy the temporal constraints imposed by that partial order.",{"name":84,"@type":75,"acceptedAnswer":85},"What are the main theoretical contributions of the work?",{"text":86,"@type":78},"It proves a Lovász-style theorem for temporal isomorphism, introduces the toadwidth measure to obtain fixed-parameter tractability for bounded toadwidth patterns, and establishes a parameterised complexity dichotomy with a tractability criterion for totally ordered temporal patterns.","https://schema.org",{"og:url":53,"og:type":89,"og:title":13,"og:site_name":60,"og:description":14},"article",{"robots":91,"canonical":53},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":94},[95,99,103,107,112,117,121,124,129,132,136],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":96,"show_sort_weight":97,"slug":98},"Story & Novel",90,"story-novel",{"id":48,"doc_module":4,"doc_module_name":47,"category_name":100,"show_sort_weight":101,"slug":102},"Literature",80,"literature",{"id":54,"doc_module":4,"doc_module_name":47,"category_name":104,"show_sort_weight":105,"slug":106},"Exam",70,"exam",{"id":108,"doc_module":4,"doc_module_name":47,"category_name":109,"show_sort_weight":110,"slug":111},5,"Comic",60,"comic",{"id":113,"doc_module":4,"doc_module_name":47,"category_name":114,"show_sort_weight":115,"slug":116},6,"Technology",50,"technology",{"id":20,"doc_module":4,"doc_module_name":47,"category_name":118,"show_sort_weight":119,"slug":120},"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":122,"slug":123},30,"research-report",{"id":125,"doc_module":4,"doc_module_name":47,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":47,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":133,"doc_module":4,"doc_module_name":47,"category_name":134,"show_sort_weight":133,"slug":135},10,"Lifestyle","lifestyle",{"id":137,"doc_module":4,"doc_module_name":47,"category_name":138,"show_sort_weight":108,"slug":139},19,"General","general"]