[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83673-en":3,"doc-seo-83673-105":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},83673,3848291630094,"Emma Wilson","https://eur-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45",8,"Research & Report","Counting Unlabeled Chordal Graphs by Equivariant Evaporation","Computes the number of unlabeled chordal graphs on n vertices, covering both total counts (OEIS A048193) and connected counts (OEIS A048192), extending previously known values beyond n=15. The approach uses a Pólya–Burnside enumeration: unlabeled counts average labeled graphs fixed by each vertex permutation. Core work evaluates π-invariant labeled chordal graphs via an equivariant dynamic program, using a divisor-bundle component decomposition and recurrences proven correct, with the full computation running in sub-exponential time and validated through n=20 by multiple independent checks.","arXiv :2607 .026 13v 1 [ cs .DM] 1 Jul 2026  \nCounting Unlabeled Chordal Graphs by Equivariant  \nEvaporation  \nMatthew Sun  \nIndependent researcher, California, USA  \n[matthewsun42@gmail.com](matthewsun42@gmail.com / mattsun1@mit.edu)[ /](matthewsun42@gmail.com / mattsun1@mit.edu)[ mattsun1@mit.edu](matthewsun42@gmail.com / mattsun1@mit.edu)  \nJune 2026  \nAbstract  \nWe compute the number of unlabeled chordal graphs on n vertices, both the total count (OEIS A048193) and the connected count (OEIS A048192), extending two sequences whose published values had remained at n = 15 . The method is a P´olya–Burnside enumeration: the number of unlabeled graphs in a class closed under relabeling is the average over Sn of the number of labeled graphs fixed by each permutation. The technical core is the evaluation, for an arbitrary permutation π, of the number of π-invariant labeled chordal graphs. We give a dynamic program for this quantity that lifts the evaporation-based labeled chordal counting of H´ebert-Johnson, Lokshtanov and Vigoda to the equivariant setting. Its central structural ingredient is a divisor-bundle decomposition: when a connected piece spans a cyclic orbit of size c, it forms, for each divisor d | c, a d-fold bundle whose constituent is an object of the same kind in the cyclic world of order c/d, computed by the same program recursively. We prove the decomposition and the correctness of the resulting recurrences, and we prove that the full Burnside computation runs in sub-exponential time nO ( √n) . We report the new terms through n = 20 and describe four independent validations, including exact agreement with all previously known values of both sequences and an Euler-transform consistency check.  \n1 Introduction  \nA graph is chordal if it has no induced cycle of length at least four. Chordal graphs are a central class in algorithmic graph theory, characterized by perfect elimination orderings and by tree decompositions whose width equals the clique number minus one. Two integer sequences count them up to isomorphism:  \nA048193 (n) = \\#{chordal graphs on n nodes},  \nA048192 (n) = \\#{connected chordal graphs on n nodes},  \nboth up to isomorphism. Their published data extended only to n = 15 .  \nFor the labeled count, H´ebert-Johnson, Lokshtanov and Vigoda [1] gave a polynomial-time dynamic program based on an evaporation process that repeatedly deletes all simplicial vertices. Counting graphs up to isomorphism, however, is governed by Burnside’s lemma and requires counting labeled chordal graphs invariant under each permutation of the vertices. H´ebert-Johnson and Lokshtanov [2] proved that counting labeled chordal graphs with a prescribed automorphism is fixed-parameter tractable in the number of moved points µ, with a running time of O(27µ n9 ); in the regime relevant to the Burnside sum, where µ may be as large as n, that bound is singleexponential in n, and no exact unlabeled enumeration was carried out there.  \nContributions. Building on the labeled evaporation dynamic program of [1], we give a concrete equivariant dynamic program for the number fix(π) of π-invariant labeled chordal graphs, and use it to compute A048192 and A048193 beyond the previous frontier. Our results are:  \n1. A divisor-bundle structure theorem (the Component-Orbit Decomposition, Theorem 1) describing how connected pieces of a π-invariant graph decompose, and reducing each equivariant recurrence to its labeled counterpart of [1] (Section 5.3) .  \n2. A correctness theorem (Theorem 3) for the assembled program, conditional only on the labeled recurrences of [1] .  \n3. A running-time theorem (Theorem 4): the entire computation of A048192(n) and A048193(n) runs in sub-exponential time nO ( √ n) = eO ( √nlog n), in contrast to the single-exponential worst case of the general parameterized bound.  \n4. The new values through n = 20 (Section 7), with four independent validations.  \n2 Preliminaries  \n2.1 The Burnside reduction  \nLet H be a class of gr","cbCainKkGjd73vhd","https://ap.wps.com/l/cbCainKkGjd73vhd","pdf",372704,4,1,12,"English","en",105,"# Abstract\n# Introduction\n## Contributions\n# Preliminaries\n## The Burnside reduction\n## Evaporation\n# The orbit-level process and its structure\n## Orbit-evaporation","[{\"question\":\"What counting problem does the document address?\",\"answer\":\"It determines the number of unlabeled chordal graphs on n vertices, including both all chordal graphs and the connected ones.\"},{\"question\":\"How are unlabeled counts reduced to a permutation-based computation?\",\"answer\":\"Using the Cauchy–Frobenius–Burnside lemma, it expresses the number of isomorphism classes as an average over permutations of labeled graphs fixed by each permutation.\"},{\"question\":\"What is the central technical method used to compute π-invariant labeled chordal graphs?\",\"answer\":\"It builds an equivariant dynamic program by extending an evaporation-based labeled counting method, supported by a divisor-bundle component (orbit) decomposition and recursively computed recurrences.\"}]",1784189659,30,{"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},"counting-unlabeled-chordal-graphs-by-equivariant-evaporation","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"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":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/counting-unlabeled-chordal-graphs-by-equivariant-evaporation/83673/",{"url":52,"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 counting problem does the document address?","Question",{"text":75,"@type":76},"It determines the number of unlabeled chordal graphs on n vertices, including both all chordal graphs and the connected ones.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are unlabeled counts reduced to a permutation-based computation?",{"text":80,"@type":76},"Using the Cauchy–Frobenius–Burnside lemma, it expresses the number of isomorphism classes as an average over permutations of labeled graphs fixed by each permutation.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the central technical method used to compute π-invariant labeled chordal graphs?",{"text":84,"@type":76},"It builds an equivariant dynamic program by extending an evaporation-based labeled counting method, supported by a divisor-bundle component (orbit) decomposition and recursively computed 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