[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82442-en":3,"doc-seo-82442-105":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},82442,7971461741311,"Ophelia","https://ap-avatar.wpscdn.com/avatar/74000253aff267980c6?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345379180704826",8,"Research & Report","Forbidding Anticomplete Planar Minors: Induced Erdős–Pósa Property and Maximum Independent Set in QP","The Erdős–Pósa theorem links absence of k disjoint cycles to the existence of a bounded vertex set whose removal eliminates all cycles. Building on Robertson–Seymour’s planar-minor extension, this work develops an induced, distance-based coarse perspective: forbidding k pairwise non-adjacent minor models of a planar graph H (kH-free graphs) can be reduced to H-minor-free by deleting bounded-size neighborhoods. The proof uses linearly many large protrusions in sparse kH-free graphs and yields logarithmic tree-width after O(log n) deletions, enabling quasi-polynomial algorithms for Maximum Independent Set.","arXiv :2607 .09646v1 [math .CO] 10 Jul 2026  \nForbidding anticomplete planar minors: Induced Erdős–Pósa property and Maximum Independent Set in QP ∗  \nMaria Chudnovsky † Amadeus Reinald ‡ Stéphan Thomassé §  \nAbstract  \nThe Erdős–Pósa theorem asserts that every graph G with no k disjoint cycles contains a set X of f (k) vertices such that G \\ X has no cycle. Robertson and Seymour showed that this Erdős–Pósa property also holds for H-minor models of any planar graph H. Equivalently, if G has no k minor models of H pairwise at distance at least 1 (i.e. disjoint), then one can remove f (k, H) balls of radius 0 (i.e. vertices) to make the graph H-minor free.  \nWe show that this coarse graph theory point of view generalizes to distance at least 2 versus radius 1 balls, yielding the induced Erdős–Pósa property for planar minors. Namely, every graph G which does not contain k pairwise non-adjacent minor models of a planar graph H (we say that G is kH-free) can be made H-minor free by removing f (k, H) neighborhoods. The proof relies on the fact that sparse kH-free graphs have linearly many independent large protrusions. The same method gives that sparse kH-free graphs can be made H-minor free by deleting O(log n) vertices (and thus have logarithmic tree-width) . This gives a quasi-polynomial algorithm for the Maximum Independent Set problem for kH-free graphs.  \n∗ This work was initiated during the ’Focused Workshop on Erdős–Pósa problems’ held in Będlewo, Poland, March 15-20, 2026, which was supported by the Banach Center of the Institute of Mathematics of the Polish Academy of Sciences.  \n†Princeton University, Princeton, NJ, USA. Supported by NSF Grants DMS-2348219 and CCF-2505100, AFOSR grant FA9550-25-1-0275, and a Guggenheim Fellowship.  \n‡University of Warsaw, Poland. Supported by Polish National Science Centre SONATA BIS-12 grant number 2022/46/E/ST6/00143 .  \n§ Univ. Lyon, ENS de Lyon, UCBL, CNRS, LIP, France. Supported by the ANR project GODASse (ANR-24-CE48-4377)  \n1 Introduction  \nThe Erdős–Pósa theorem [18] asserts that every graph G with no k disjoint cycles contains a set X of f (k) vertices such that G \\ X is a forest. In other words, the minimum feedback vertex set is functionally equivalent to the maximum packing of cycles. This bridged duality gap is referred to asthe Erdős–Pósa property (EPP), and the term is now used for a vast collection of analogous results relating the packing and hitting number of more general objects. It often has direct algorithmic applications. For instance, Maximum Independent Set (MIS) is polynomial-time solvable in graphs without k disjoint cycles, since their feedback vertex set, and hence their tree-width, is bounded.  \nTo this day, variants of cycles have constituted a rich playing ground for the Erdős–Pósa property, in terms of both positive and negative results. On the positive side, the EPP is known to hold for cycles with length constraints [42, 32], holes [28], cycles passing through prescribed vertices [27, 37], and even oriented cycles [39] . On the negative side, odd cycles in an Escher Wall pairwise intersect yet their hitting set is unbounded [38], induced cycles of length at least 5 do not have the EPP [28], and neither do cycles of prime lengths [24] . If we wish to generalize the EPP to more complex objects, one can observe that cycles are K3-minor models. It is then natural to ask if minor models of some fixed graph H have the EPP. This is not always the case since toroidal grids do not contain two disjoint K5-minor models, whereas the minimum hitting set of K5-minor models has unbounded size. Arguably, the most striking result in this field is that H-minor models have the EPP if and only if H is planar. This is a direct application of Robertson and Seymour’s celebrated grid minor theorem [40]: either a graph has a large grid minor, and it contains k disjoint H-minor models, or it has bounded tree-width, in which case a simple divide and conquer argument gives a hitting","cbCaib626aqd0RvJ","https://ap.wps.com/l/cbCaib626aqd0RvJ","pdf",530744,2,1,21,"English","en",105,"# Abstract\n# Introduction","[{\"question\":\"What does the classic Erdős–Pósa theorem guarantee for graphs without k disjoint cycles?\",\"answer\":\"It guarantees a function f(k) such that there exists a set X of at most f(k) vertices whose removal makes the graph cycle-free (equivalently, the remainder has no cycles).\"},{\"question\":\"How does this work extend Erdős–Pósa theory from disjointness to an induced, distance-based setting for planar minors?\",\"answer\":\"It shows that for a planar graph H, if a graph G is kH-free—meaning it has no k pairwise non-adjacent minor models of H—then deleting f(k,H) neighborhoods makes G H-minor-free. The approach converts a coarse distance/non-adjacency constraint into a bounded “hitting” set using graph structure (protrusions).\"},{\"question\":\"What algorithmic consequences does the protrusion method give for Maximum Independent Set on kH-free graphs?\",\"answer\":\"The method implies that sparse kH-free graphs can be made H-minor-free by deleting O(log n) vertices, yielding logarithmic tree-width, which supports a quasi-polynomial algorithm for Maximum Independent Set on kH-free graphs.\"}]",1784180396,53,{"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},"forbidding-anticomplete-planar-minors-induced-erdosposa-property-and-maximum-independent-set-in-qp","",{"@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/forbidding-anticomplete-planar-minors-induced-erdosposa-property-and-maximum-independent-set-in-qp/82442/",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","2026-07-22","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 classic Erdős–Pósa theorem guarantee for graphs without k disjoint cycles?","Question",{"text":75,"@type":76},"It guarantees a function f(k) such that there exists a set X of at most f(k) vertices whose removal makes the graph cycle-free (equivalently, the remainder has no cycles).","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does this work extend Erdős–Pósa theory from disjointness to an induced, distance-based setting for planar minors?",{"text":80,"@type":76},"It shows that for a planar graph H, if a graph G is kH-free—meaning it has no k pairwise non-adjacent minor models of H—then deleting f(k,H) neighborhoods makes G H-minor-free. The approach converts a coarse distance/non-adjacency constraint into a bounded “hitting” set using graph structure (protrusions).",{"name":82,"@type":73,"acceptedAnswer":83},"What algorithmic consequences does the protrusion method give for Maximum Independent Set on kH-free graphs?",{"text":84,"@type":76},"The method implies that sparse kH-free graphs can be made H-minor-free by deleting O(log n) vertices, yielding logarithmic tree-width, which supports a quasi-polynomial algorithm for Maximum Independent Set on kH-free graphs.","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"]