[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82324-en":3,"doc-seo-82324-105":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},82324,1374391974564,"Clementine","https://ap-avatar.wpscdn.com/avatar/14000253aa45c000a9e?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779874745381141002",8,"Research & Report","Complexity of the Graph Homomorphism Problem w.r.t. Degeneracy","Graph Homomorphism Problem (HOM) asks whether a source graph G with n vertices admits an edge-preserving mapping into a target graph H with h vertices. While brute force runs in O(h^n) and ETH rules out 2^{o(n log h)}-type improvements, parameterized approaches solve HOM in p(H)·O(n) for parameters such as treewidth or maximum degree. This work analyzes HOM under the degeneracy of H, proving ETH-based lower bounds: no algorithm of type 2^{o(degen(H)·n)} exists for any degeneracy as a function of n. Bounded degeneracy alone does not yield benign target size, and a no-compression barrier limits tightening fine-grained bounds under ETH.","arXiv :2607 .09377v 1 [ cs .CC] 10 Jul 2026  \nComplexity of the Graph Homomorphism Problem w.r.t. Degeneracy Grigorii Braulov ∗ Nikolai Chukhin † Alexander S. Kulikov ‡ Ivan Mihajlin §  \nAbstract  \nThe graph homomorphism problem HOM is: given an n-vertex source graph G and an h-vertex target graph H, is there a mapping from V (G) to V (H) that preserves edges? A straightforward brute-force algorithm for HOM has running time O(hn ) = O(2nlog h ) and it is known that, under ETH, there are no 2o (nlog h) algorithms. In recent years, less restrictive graph parameters p have been identified that allow one to solve HOM in time p(H)O (n) . Examples include treewidth, maximum degree, and track number. These algorithms are faster than O (hn ) and allow one to solve HOM in plain-exponential time 2O (n) in the special case when p(H) is bounded. On the other hand, it is known that the chromatic number parameter is too small: under ETH, HOM cannot be solved in time χ(H)O (n) .  \nWe study the complexity of HOM in terms of the degeneracy of H. This is perhaps the most natural unresolved graph parameter between the known algorithmic and hardness regimes: on the one hand, each of bounded treewidth, bounded maximum degree, and bounded track number implies bounded degeneracy; on the other hand, bounded degeneracy implies bounded chromatic number. Our results show that, at the same time, the influence of degeneracy of H on the complexity of HOM differs significantly from that of the previously studied parameters. We show that, under ETH, there is no 2o(degen(H)n) algorithm for any value of degen(H) as a function of n. We also show that bounded degeneracy alone does not make target size benign: even targets with degen(H) ≤ 2 and quasi-polynomial size force nΩ(n)-scale hardness. Finally, we introduce a no-compression barrier that explains why the known fine-grained lower bounds for sparse 2-CSP are not tight under ETH. Moreover, it shows that substantially stronger lower bounds for polynomial-target degeneracy are unlikely to follow from standard reductions from sparse 3-SAT.  \n∗ Neapolis University Pafos. Email: [braulov2004@gmail.com](braulov2004@gmail.com)  \n†JetBrains Research. Email: [buyolitsez1951@gmail.com](buyolitsez1951@gmail.com)  \n‡JetBrains Research. Email: [alexander.s.kulikov@gmail.com](alexander.s.kulikov@gmail.com)  \n§ JetBrains Research. Email: [ivmihajlin@gmail.com](ivmihajlin@gmail.com)  \nContents  \n1 Graph Homomorphism and Target Parameters 3  \n1. 1 Our Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4  \n2 Preliminaries 6  \n2.1 Homomorphism Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6  \n2.2 Graph Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7  \n2.3 Constraint Satisfaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7  \n2.4 Boolean Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7  \n2.5 SAT Complexity Hypotheses ............................... 8  \n3 Linear Dependence on Degeneracy 8  \n3. 1 Proof Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9  \n3.2 Bucket Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9  \n3.3 List-Homomorphism Instance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11  \n3.4 Removing the Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16  \n3.5 Finishing the Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21  \n4 The Degeneracy Threshold from One to Two 23  \n4.1 Sparse Pinning Frame ................................... 23  \n4.2 Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25  \n5 No-Compression Barriers for SAT Reductions 28  \n5.1 Sparse-SAT Information Scale ............................... 28  \n5.2 Application to Sparse CSP ....................","cbCaijn3eH6HqiDX","https://ap.wps.com/l/cbCaijn3eH6HqiDX","pdf",757066,1,53,"English","en",105,"# Graph Homomorphism and Target Parameters\n## Our Results\n# Preliminaries\n## Homomorphism Problems\n## Graph Parameters\n## Constraint Satisfaction\n## Boolean Circuits\n## SAT Complexity Hypotheses\n# Linear Dependence on Degeneracy\n## Proof Overview\n## Bucket Decomposition\n## List-Homomorphism Instance\n## Removing the Lists\n## Finishing the Lower Bound\n# The Degeneracy Threshold from One to Two\n## Sparse Pinning Frame\n## Reduction\n# No-Compression Barriers for SAT Reductions\n## Sparse-SAT Information Scale\n## Application to Sparse CSP\n## Succinct List-Extension Homomorphism Problem\n## Reduction\n## No-Compression Barrier\n# Generated Formulas and Hard Witnesses\n# Open Problems\n# Bounded Chromatic Number\n# Graph Parameters\n# The Graph Homomorphism Problem and Its Complexity Dichotomies\n# No-Compression Barrier (SETH Version)","[{\"question\":\"What does the Graph Homomorphism Problem (HOM) ask?\",\"answer\":\"Given graphs G and H, HOM asks whether there exists a mapping from V(G) to V(H) that preserves edges.\"},{\"question\":\"How do earlier complexity results characterize HOM beyond brute force?\",\"answer\":\"Brute force runs in O(h^n), and under ETH there are no 2^{o(n log h)} algorithms. Parameterized algorithms using parameters like treewidth or maximum degree can yield p(H)·O(n) time, and when p(H) is bounded this becomes 2^{O(n)}.\"},{\"question\":\"What is the main impact of target degeneracy on HOM complexity in this work?\",\"answer\":\"The paper proves an ETH-based lower bound showing there is no 2^{o(degen(H)·n)} algorithm for any degeneracy value as a function of n. It also shows bounded degeneracy (even up to degen(H) ≤ 2) does not make the target size benign, leading to n^{Ω(n)}-scale hardness in certain regimes.\"}]",1784179632,134,{"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},"complexity-of-the-graph-homomorphism-problem-wrt-degeneracy","",{"@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/complexity-of-the-graph-homomorphism-problem-wrt-degeneracy/82324/",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 the Graph Homomorphism Problem (HOM) ask?","Question",{"text":75,"@type":76},"Given graphs G and H, HOM asks whether there exists a mapping from V(G) to V(H) that preserves edges.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do earlier complexity results characterize HOM beyond brute force?",{"text":80,"@type":76},"Brute force runs in O(h^n), and under ETH there are no 2^{o(n log h)} algorithms. Parameterized algorithms using parameters like treewidth or maximum degree can yield p(H)·O(n) time, and when p(H) is bounded this becomes 2^{O(n)}.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the main impact of target degeneracy on HOM complexity in this work?",{"text":84,"@type":76},"The paper proves an ETH-based lower bound showing there is no 2^{o(degen(H)·n)} algorithm for any degeneracy value as a function of n. It also shows bounded degeneracy (even up to degen(H) ≤ 2) does not make the target size benign, leading to n^{Ω(n)}-scale hardness in certain regimes.","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":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]