[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84262-en":3,"doc-seo-84262-105":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},84262,1374391974564,"Clementine","https://ap-avatar.wpscdn.com/avatar/14000253aa45c000a9e?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779874745381141002",8,"Research & Report","Induced Erdős–Pósa Property for Long Holes, Long Thetas, and Beyond","Induced Erdős–Pósa properties connect maximum packing of pairwise anti-adjacent copies of a graph object with the minimum size of a neighborhood set needed to hit all copies. This work proves that for any fixed length t, long cycles Ct and long thetas Θt have the induced Erdős–Pósa property under the induced minor relation. For every k and graph G, either G contains k vertex-disjoint anti-adjacent induced-minor copies, or a set X of size O(tk log k) whose neighborhood N[X] hits all such induced minors. The results yield balanced separators and a QPTAS for Maximum Weight Independent Set and extensions.","arXiv :2607 .07697v1 [math .CO] 8 Jul 2026  \nInduced Erdős–Pósa property for long holes, long thetas, and beyond  \nJadwiga Czyżewska∗ Tomáš Masařík† Marcin Pilipczuk‡ Amadeus Reinald§  \nPaweł Rzążewski¶  \nAbstract  \nThe induced Erdős–Pósa property in graphs relates the maximum packing of pairwise anti-adjacent copies of an object with the minimal number of neighborhoods required to hit all copies. In this paper, the objects we consider are long cycles and long thetas, both as induced minors. Let Ct denote the cycle with t vertices and let Θt be the graph consisting of three internally disjoint and anti-adjacent paths, each with t internal vertices, connecting the same pair of distinct vertices.  \nWe show that for every fixed t, both Ct and Θt have the induced Erdős–Pósa property with respect to the induced minor relation. More precisely, for every integer k and a graph G, one of the two outcomes occurs:  \n• G contains k pairwise vertex-disjoint and anti-adjacent copies of Ct (resp., Θt) as induced minors, or  \n• there is some X ⊆ V (G) of size O(tk log k) such that the set N[X], consisting of X and its neighbors, hits all Ct (resp., all Θt) induced minors in G.  \nThis resolves in a strong form a special case of a conjecture of Ahn, Gollin, Huynh, and Kwon [SODA 2025] . From these results we derive that graphs that exclude kΘt as an induced minor admit balanced separators  \nconsisting of the neighborhood of O(tk log k) vertices. This in turn resolves a special case of a conjecture of Gartland and Lokshtanov and, combined with known techniques, yields a QPTAS for Maximum Weight Independent Set and a number of its generalizations.  \n∗University of Warsaw, Poland ([j.czyzewska@mimuw.edu.pl](j.czyzewska@mimuw.edu.pl)). Supported by Polish National Science Centre SONATA BIS-12 grant number 2022/46/E/ST6/00143 .  \n†University of Warsaw, Poland ([masarik@mimuw.edu.pl](masarik@mimuw.edu.pl)) . Supported by the Polish National Science Centre SONATA-17 grant number 2021/43/D/ST6/03312 .  \n‡University of Warsaw, Poland ([m.pilipczuk@mimuw.edu.pl](m.pilipczuk@mimuw.edu.pl)) . Supported by Polish National Science Centre SONATA BIS-12 grant number 2022/46/E/ST6/00143 .  \n§University of Warsaw, Poland ([reinald@mimuw.edu.pl](reinald@mimuw.edu.pl)) . Supported by Polish National Science Centre SONATA BIS-12 grant number 2022/46/E/ST6/00143 .  \n¶ Warsaw University of Technology, Poland ([pawel.rzazewski@pw.edu.pl](pawel.rzazewski@pw.edu.pl)) . Supported by the National Science Centre grant 2024/54/E/ST6/00094 .  \nContents  \n1 Introduction 1  \n2 Models 5  \n2. 1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5  \n2.2 Ear addition and ear-decompositions .................................. 7  \n2.3 Properties ................................................. 9  \n3 Algorithmic induced Erdős–Pósa property for long holes 10  \n4 Preliminaries 13  \n4.1 Long thetas and long three-path-configurations ............................ 13  \n5 Obtaining long 3PCs from (sub)cubic models 14  \n5.1 Thetas in models give long 3PCs ..................................... 14  \n5.2 Packing thetas in a subcubic model ................................... 15  \n6 Algorithmic induced Erdős–Pósa property for long 3PCs 16  \n6.1 Long 3PCs and models of Θ0 ....................................... 17  \n6. 2 Initial model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18  \n6.3 Proof of Theorem 1 .6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20  \n7 Induced Erdős–Pósa property for long thetas induced minors 22  \n7.1 Big component and measure ....................................... 23  \n7.2 Finding a short t-model of Θ0 ...................................... 24  \n7.3 Proof of Theorem 1 .5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26  \n8 From induced Erdős–Pósa property to dominated balanced separ","cbCaicHPPuEMBcxk","https://ap.wps.com/l/cbCaicHPPuEMBcxk","pdf",840670,3,1,35,"English","en",105,"# Abstract\n# Introduction\n# Models\n## Definitions\n## Ear addition and ear-decompositions\n## Properties\n# Algorithmic induced Erdős–Pósa property for long holes\n# Preliminaries\n## Long thetas and long three-path-configurations\n# Obtaining long 3PCs from (sub)cubic models\n## Thetas in models give long 3PCs\n## Packing thetas in a subcubic model\n# Algorithmic induced Erdős–Pósa property for long 3PCs\n## Long 3PCs and models of Θ0\n## Initial model\n## Proof of Theorem 1.6\n# Induced Erdős–Pósa property for long thetas induced minors\n## Big component and measure\n## Finding a short t-model of Θ0\n## Proof of Theorem 1.5\n# From induced Erdős–Pósa property to dominated balanced separators\n# Long thetas in blob graphs\n# Conclusion and further work","[{\"question\":\"What induced Erdős–Pósa outcomes does the paper establish for long cycles Ct and long thetas Θt?\",\"answer\":\"For each fixed t, every graph G and integer k satisfy one of two cases: either G contains k pairwise vertex-disjoint anti-adjacent induced-minor copies of Ct (or Θt), or there exists X⊆V(G) with |X|=O(tk log k) such that the neighborhood N[X] hits all induced-minor copies.\"},{\"question\":\"Which graph relation and object types are studied in the induced Erdős–Pósa property?\",\"answer\":\"The paper studies the induced Erdős–Pósa property with respect to the induced minor relation, focusing on long cycles Ct and long thetas Θt as the packed objects.\"},{\"question\":\"How do these induced Erdős–Pósa results impact separator theorems and algorithmic problems?\",\"answer\":\"They imply that graphs excluding kΘt as an induced minor admit balanced separators formed from the neighborhood of O(tk log k) vertices, and these tools lead to a QPTAS for Maximum Weight Independent Set and related generalizations.\"}]",1784194461,88,{"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},"induced-erdosposa-property-for-long-holes-long-thetas-and-beyond","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"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":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/induced-erdosposa-property-for-long-holes-long-thetas-and-beyond/84262/",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","2026-07-25","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 induced Erdős–Pósa outcomes does the paper establish for long cycles Ct and long thetas Θt?","Question",{"text":75,"@type":76},"For each fixed t, every graph G and integer k satisfy one of two cases: either G contains k pairwise vertex-disjoint anti-adjacent induced-minor copies of Ct (or Θt), or there exists X⊆V(G) with |X|=O(tk log k) such that the neighborhood N[X] hits all induced-minor copies.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which graph relation and object types are studied in the induced Erdős–Pósa property?",{"text":80,"@type":76},"The paper studies the induced Erdős–Pósa property with respect to the induced minor relation, focusing on long cycles Ct and long thetas Θt as the packed objects.",{"name":82,"@type":73,"acceptedAnswer":83},"How do these induced Erdős–Pósa results impact separator theorems and algorithmic problems?",{"text":84,"@type":76},"They imply that graphs excluding kΘt as an induced minor admit balanced separators formed from the neighborhood of O(tk log k) vertices, and these tools lead to a QPTAS for Maximum Weight Independent Set and related generalizations.","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":47,"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"]