[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84688-en":3,"doc-seo-84688-105":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},84688,4398048949847,"Eliana","https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267",8,"Research & Report","Oddomorphisms, Split-Off Minors, and the Strong Roberson Conjecture","Existence of an oddomorphism from a graph F to a graph G does not guarantee that G is a minor of F. This settles a question of Roberson (2022) and shows that CFI graphs cannot directly be used to prove the Strong Roberson Conjecture. The work defines split-off minors and proves that an oddomorphism from F to G implies G is a split-off minor of F. Classes closed under split-off minors and disjoint unions become homomorphism distinguishing closed, advancing links between graph containment and homomorphism indistinguishability.","arXiv :2607 .03405v 1 [ cs .DM] 3 Jul 2026  \nOddomorphisms, Split-Off Minors, and the Strong Roberson  \nConjecture  \nArnar Á . Kristjánsson  \nJuly 7, 2026  \nAbstract  \nWe show that the existence of an oddomorphism from a graph F to a graph G does not imply that G is a minor of F. This answers a question posed by Roberson (2022) and shows that the CFI graphs cannot be used to prove the Strong Roberson Conjecture. Additionally, we introduce the concept of a split-off minor and show that the existence of an oddomorphism from F to G implies that G is a split-off minor of F. Consequently, every class that is closed undertaking split-off minors and disjoint unions is homomorphism distinguishing closed. The split-off minor relation is the first minor-like structural relation shown to have this property, marking a meaningful advancement in our understanding of the interaction between structural graph containment and homomorphism indistinguishability relations.  \n1 Introduction  \nA central technique in the study of limitative results and lower bounds in theoretical computer science is the construction of a pair of distinct structures that appear nearly identical. To this end, the Cai–Fürer–Immerman (CFI) construction has established itself as an exceptionally versatile tool. It produces, from a given base graph G, a pair of graphs CFI0 (G), CFI 1 (G), which are locally similar but non-isomorphic. The construction was introduced by Cai, Fürer, and Immerman [1] to show that, for any k , k-variable counting logic cannot distinguish all non-isomorphic graphs. Equivalently, this demonstrates that the k − 1-dimensional Weisfeiler-Leman algorithm fails to solve the graph isomorphism problem. Subsequently, the construction has been adapted in numerous forms across a wide range of problems, becoming ubiquitous in finite model theory. For example, variations of it have repeatedly been used to separate logics from PTIME [4, 14 , 18] and in the analysis of the Weisfeiler-Leman algorithm [11, 10 , 16] . Recently, it has also been used to establish a \\#P-hardness result [2] and to prove the equirank homomorphism preservation theorem [27] .  \nAnother line of research that has been gaining traction in recent years is the use of homomorphism counts for characterizing relations between structures. The foundational result by Lovász [19] states that two finite graphs are isomorphic if and only if they have the same number of homomorphisms from all finite graphs. Dvořak [8] later proved that a pair of graphs is indistinguishable in k-variable counting logic if and only if they have the same number of homomorphisms from every graph of treewidth at most k − 1, and Grohe [15] proved an analogous result relating counting logic with quantifier rank bounded by k to homomorphism counts from graphs of treedepth at most k − 1. These results were generalized categorically by Dawar, Jakl, and Reggio [3] and are now often referred to as Lovász-type theorems. Another important Lovász-type theorem that does not fit into the categorical generalisation is the result by Mančinska and Roberson [20] that a pair of graphs is  \nquantum isomorphic if and only if they have the same number of homomorphisms from all planar graphs.  \nThese results led to the study of homomorphism indistinguishability relations more generally. For a class F of graphs, its homomorphism indistinguishability relation ≡F consists of the pairs of graphs that have the same homomorphism count from every element of F. Roberson [23] posed the question of when two such relations ≡F1 , ≡F2 are distinct. He observed that if F1 and F2 are the maximal classes defining their homomorphism indistinguishability relation, the comparison between the indistinguishability relations becomes equivalent to the comparison between the underlying classes of graphs. If a class F satisfies this maximality condition, we say that it is homomorphism distinguishing closed, abbreviated h. d. closed. Roberson made the following conjec","cbCaihewcrHoUqrx","https://ap.wps.com/l/cbCaihewcrHoUqrx","pdf",514636,1,22,"English","en",105,"# Abstract\n# Introduction\n## CFI construction and logic separation\n## Homomorphism indistinguishability and Lovász-type theorems\n## Roberson conjectures: strong and weak\n## Oddomorphisms and their role with CFI graphs\n## Main theorem connecting homomorphisms and weak oddomorphisms","[{\"question\":\"What does the paper show about oddomorphisms and graph minors?\",\"answer\":\"An oddomorphism from a graph F to a graph G does not imply that G is a minor of F. This answers a question by Roberson (2022).\"},{\"question\":\"Why are CFI graphs relevant to the Strong Roberson Conjecture in this work?\",\"answer\":\"The results show that CFI graphs cannot be used in the intended way to prove the Strong Roberson Conjecture, because oddomorphism existence does not force minor containment.\"},{\"question\":\"What is a split-off minor, and how does it relate to oddomorphisms?\",\"answer\":\"The paper introduces split-off minors and proves that if there exists an oddomorphism from F to G, then G is a split-off minor of F. This leads to closure results for homomorphism distinguishing properties.\"}]",1784197671,55,{"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},"oddomorphisms-split-off-minors-and-the-strong-roberson-conjecture","",{"@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/oddomorphisms-split-off-minors-and-the-strong-roberson-conjecture/84688/",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 paper show about oddomorphisms and graph minors?","Question",{"text":75,"@type":76},"An oddomorphism from a graph F to a graph G does not imply that G is a minor of F. This answers a question by Roberson (2022).","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why are CFI graphs relevant to the Strong Roberson Conjecture in this work?",{"text":80,"@type":76},"The results show that CFI graphs cannot be used in the intended way to prove the Strong Roberson Conjecture, because oddomorphism existence does not force minor containment.",{"name":82,"@type":73,"acceptedAnswer":83},"What is a split-off minor, and how does it relate to oddomorphisms?",{"text":84,"@type":76},"The paper introduces split-off minors and proves that if there exists an oddomorphism from F to G, then G is a split-off minor of F. 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