Symbolic Rees algebras, vertex covers and irreducible representations of Rees cones
Let \(G\) be a simple graph and let \(I_c(G)\) be its ideal of vertex covers. We give a graph theoretical description of the irreducible \(b\)-vertex covers of \(G\), i.e., we describe the minimal generators of the symbolic Rees algebra of \(I_c(G)\). Then we study the irreducible \(b\)-vertex cover...
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| Дата: | 2018 |
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| Формат: | Стаття |
| Мова: | English |
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Lugansk National Taras Shevchenko University
2018
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/651 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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oai:ojs.admjournal.luguniv.edu.ua:article-6512018-04-04T09:17:05Z Symbolic Rees algebras, vertex covers and irreducible representations of Rees cones Dupont, Luis A. Villarreal, Rafael H. edge ideal, symbolic Rees algebras, perfect graph, irreducible vertex covers, irreducible graph, Alexander dual, blocker, clutter 13F20, 05C75, 05C65, 52B20 Let \(G\) be a simple graph and let \(I_c(G)\) be its ideal of vertex covers. We give a graph theoretical description of the irreducible \(b\)-vertex covers of \(G\), i.e., we describe the minimal generators of the symbolic Rees algebra of \(I_c(G)\). Then we study the irreducible \(b\)-vertex covers of the blocker of \(G\), i.e., we study the minimal generators of the symbolic Rees algebra of the edge ideal of \(G\). We give a graph theoretical description of the irreducible binary \(b\)-vertex covers of the blocker of \(G\). It is shown that they correspond to irreducible induced subgraphs of \(G\). As a byproduct we obtain a method, using Hilbert bases, to obtain all irreducible induced subgraphs of \(G\). In particular we obtain all odd holes and antiholes. We study irreducible graphs and give a method to construct irreducible \(b\)-vertex covers of the blocker of \(G\) with high degree relative to the number of vertices of \(G\). Lugansk National Taras Shevchenko University 2018-04-04 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/651 Algebra and Discrete Mathematics; Vol 10, No 2 (2010) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/651/185 Copyright (c) 2018 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| collection |
OJS |
| language |
English |
| topic |
edge ideal symbolic Rees algebras perfect graph irreducible vertex covers irreducible graph Alexander dual blocker clutter 13F20 05C75 05C65 52B20 |
| spellingShingle |
edge ideal symbolic Rees algebras perfect graph irreducible vertex covers irreducible graph Alexander dual blocker clutter 13F20 05C75 05C65 52B20 Dupont, Luis A. Villarreal, Rafael H. Symbolic Rees algebras, vertex covers and irreducible representations of Rees cones |
| topic_facet |
edge ideal symbolic Rees algebras perfect graph irreducible vertex covers irreducible graph Alexander dual blocker clutter 13F20 05C75 05C65 52B20 |
| format |
Article |
| author |
Dupont, Luis A. Villarreal, Rafael H. |
| author_facet |
Dupont, Luis A. Villarreal, Rafael H. |
| author_sort |
Dupont, Luis A. |
| title |
Symbolic Rees algebras, vertex covers and irreducible representations of Rees cones |
| title_short |
Symbolic Rees algebras, vertex covers and irreducible representations of Rees cones |
| title_full |
Symbolic Rees algebras, vertex covers and irreducible representations of Rees cones |
| title_fullStr |
Symbolic Rees algebras, vertex covers and irreducible representations of Rees cones |
| title_full_unstemmed |
Symbolic Rees algebras, vertex covers and irreducible representations of Rees cones |
| title_sort |
symbolic rees algebras, vertex covers and irreducible representations of rees cones |
| description |
Let \(G\) be a simple graph and let \(I_c(G)\) be its ideal of vertex covers. We give a graph theoretical description of the irreducible \(b\)-vertex covers of \(G\), i.e., we describe the minimal generators of the symbolic Rees algebra of \(I_c(G)\). Then we study the irreducible \(b\)-vertex covers of the blocker of \(G\), i.e., we study the minimal generators of the symbolic Rees algebra of the edge ideal of \(G\). We give a graph theoretical description of the irreducible binary \(b\)-vertex covers of the blocker of \(G\). It is shown that they correspond to irreducible induced subgraphs of \(G\). As a byproduct we obtain a method, using Hilbert bases, to obtain all irreducible induced subgraphs of \(G\). In particular we obtain all odd holes and antiholes. We study irreducible graphs and give a method to construct irreducible \(b\)-vertex covers of the blocker of \(G\) with high degree relative to the number of vertices of \(G\). |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/651 |
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AT dupontluisa symbolicreesalgebrasvertexcoversandirreduciblerepresentationsofreescones AT villarrealrafaelh symbolicreesalgebrasvertexcoversandirreduciblerepresentationsofreescones |
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2024-04-12T06:26:34Z |
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2024-04-12T06:26:34Z |
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