Irreducible classes and barycentric subdivision on triangle-free 3 connected matroids

Detalhes bibliográficos
Autor(a) principal: SANTOS FILHO, Jaime Cesar dos
Data de Publicação: 2020
Tipo de documento: Tese
Idioma: por
Título da fonte: Repositório Institucional da UFPE
dARK ID: ark:/64986/0013000012t16
Texto Completo: https://repositorio.ufpe.br/handle/123456789/39037
Resumo: The 3-connected matroids, fundamental in matroid theory, have two families of irreducible matroids with respect to the operations of deletion and contraction. This result is known as Tutte’s Wheels and Whirls Theorem, established in [11]. Lemos, in [4], considered seven reduction operations to classify the triangles-free 3-connected matroids, five in addition to the two considered by Tutte. The results obtained by Lemos generalize those obtained by Kriesell [2]. Considering only the first three reduction operations defined in [4], we prove that 4 local structures formed by squares and triads behave like "building blocks" for these families of irreducible. Subdividing the seventh reduction, we add another family of triangle-free 3-connected matoids: diamantic matroids. We have established, in a constructive way, that for each matroid in this family there is a unique totally triangular matoid associated. The construction of this one-to-one correspondence is based on the generalized parallel connection and passes through a matroid, unique up to isomorphisms, which corresponds to the barycentric subdivision in the case of graphic matroids.
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spelling SANTOS FILHO, Jaime Cesar doshttp://lattes.cnpq.br/1522562369123416http://lattes.cnpq.br/2150972086881898LEMOS, Manoel José Machado Soares2021-01-12T19:16:18Z2021-01-12T19:16:18Z2020-01-30SANTOS FILHO, Jaime Cesar dos. Irreducible classes and barycentric subdivision on triangle-free 3 connected matroids. 2020. Tese (Doutorado em Matemática) - Universidade Federal de Pernambuco, Recife, 2020.https://repositorio.ufpe.br/handle/123456789/39037ark:/64986/0013000012t16The 3-connected matroids, fundamental in matroid theory, have two families of irreducible matroids with respect to the operations of deletion and contraction. This result is known as Tutte’s Wheels and Whirls Theorem, established in [11]. Lemos, in [4], considered seven reduction operations to classify the triangles-free 3-connected matroids, five in addition to the two considered by Tutte. The results obtained by Lemos generalize those obtained by Kriesell [2]. Considering only the first three reduction operations defined in [4], we prove that 4 local structures formed by squares and triads behave like "building blocks" for these families of irreducible. Subdividing the seventh reduction, we add another family of triangle-free 3-connected matoids: diamantic matroids. We have established, in a constructive way, that for each matroid in this family there is a unique totally triangular matoid associated. The construction of this one-to-one correspondence is based on the generalized parallel connection and passes through a matroid, unique up to isomorphisms, which corresponds to the barycentric subdivision in the case of graphic matroids.As matroides 3-conexas, fundamentais na teoria das matroides, possuem duas família de irredutíveis com relação às operações de deleção e contração. Este resultado é conhecido como Teorema da Roda e do Redemoinho de Tutte [11]. Lemos, em [4], considerou sete operações de redução para classificar as matroides 3-conexas livre de triângulos irredutíveis, cinco além das duas consideradas por Tutte. Os resultados obtidos por Lemos generalizam os obtidos por Kriesell [2]. Considerando apenas as três primeiras operações de redução definidas em [4], provamos que 4 estruturas locais formadas por quadrados e triades se comportam como "blocos construtores" para estas famílias de irredutíveis. Subdividindo a sétima redução, acrescentamos mais uma família de matroides 3-conexas livre de triângulos irredutíveís: diamantic matroids, em inglês. Estabelecemos, de uma forma construtiva, que para cada matroide nesta família existe um única matroide totalmente triangular associada. A construção desta correspondência biunívoca é baseada na conexão em paralelo generalizada e passa por uma matroide, única a menos de isomorfismos, que corresponde a subdivisão baricêntrica no caso de matroides gráficas.porUniversidade Federal de PernambucoPrograma de Pos Graduacao em MatematicaUFPEBrasilAttribution-NonCommercial-NoDerivs 3.0 Brazilhttp://creativecommons.org/licenses/by-nc-nd/3.0/br/info:eu-repo/semantics/openAccessCombinatóriaMatroidesIrreducible classes and barycentric subdivision on triangle-free 3 connected matroidsinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesisdoutoradoreponame:Repositório Institucional da UFPEinstname:Universidade Federal de Pernambuco (UFPE)instacron:UFPECC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8811https://repositorio.ufpe.br/bitstream/123456789/39037/2/license_rdfe39d27027a6cc9cb039ad269a5db8e34MD52LICENSElicense.txtlicense.txttext/plain; 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dc.title.pt_BR.fl_str_mv Irreducible classes and barycentric subdivision on triangle-free 3 connected matroids
title Irreducible classes and barycentric subdivision on triangle-free 3 connected matroids
spellingShingle Irreducible classes and barycentric subdivision on triangle-free 3 connected matroids
SANTOS FILHO, Jaime Cesar dos
Combinatória
Matroides
title_short Irreducible classes and barycentric subdivision on triangle-free 3 connected matroids
title_full Irreducible classes and barycentric subdivision on triangle-free 3 connected matroids
title_fullStr Irreducible classes and barycentric subdivision on triangle-free 3 connected matroids
title_full_unstemmed Irreducible classes and barycentric subdivision on triangle-free 3 connected matroids
title_sort Irreducible classes and barycentric subdivision on triangle-free 3 connected matroids
author SANTOS FILHO, Jaime Cesar dos
author_facet SANTOS FILHO, Jaime Cesar dos
author_role author
dc.contributor.authorLattes.pt_BR.fl_str_mv http://lattes.cnpq.br/1522562369123416
dc.contributor.advisorLattes.pt_BR.fl_str_mv http://lattes.cnpq.br/2150972086881898
dc.contributor.author.fl_str_mv SANTOS FILHO, Jaime Cesar dos
dc.contributor.advisor1.fl_str_mv LEMOS, Manoel José Machado Soares
contributor_str_mv LEMOS, Manoel José Machado Soares
dc.subject.por.fl_str_mv Combinatória
Matroides
topic Combinatória
Matroides
description The 3-connected matroids, fundamental in matroid theory, have two families of irreducible matroids with respect to the operations of deletion and contraction. This result is known as Tutte’s Wheels and Whirls Theorem, established in [11]. Lemos, in [4], considered seven reduction operations to classify the triangles-free 3-connected matroids, five in addition to the two considered by Tutte. The results obtained by Lemos generalize those obtained by Kriesell [2]. Considering only the first three reduction operations defined in [4], we prove that 4 local structures formed by squares and triads behave like "building blocks" for these families of irreducible. Subdividing the seventh reduction, we add another family of triangle-free 3-connected matoids: diamantic matroids. We have established, in a constructive way, that for each matroid in this family there is a unique totally triangular matoid associated. The construction of this one-to-one correspondence is based on the generalized parallel connection and passes through a matroid, unique up to isomorphisms, which corresponds to the barycentric subdivision in the case of graphic matroids.
publishDate 2020
dc.date.issued.fl_str_mv 2020-01-30
dc.date.accessioned.fl_str_mv 2021-01-12T19:16:18Z
dc.date.available.fl_str_mv 2021-01-12T19:16:18Z
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dc.identifier.citation.fl_str_mv SANTOS FILHO, Jaime Cesar dos. Irreducible classes and barycentric subdivision on triangle-free 3 connected matroids. 2020. Tese (Doutorado em Matemática) - Universidade Federal de Pernambuco, Recife, 2020.
dc.identifier.uri.fl_str_mv https://repositorio.ufpe.br/handle/123456789/39037
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identifier_str_mv SANTOS FILHO, Jaime Cesar dos. Irreducible classes and barycentric subdivision on triangle-free 3 connected matroids. 2020. Tese (Doutorado em Matemática) - Universidade Federal de Pernambuco, Recife, 2020.
ark:/64986/0013000012t16
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