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arXiv:0712.1249 (math)
[Submitted on 8 Dec 2007 (v1), last revised 4 Mar 2009 (this version, v3)]

Title:Symbolic Rees algebras, vertex covers and irreducible representations of Rees cones

Authors:Luis A. Dupont, Rafael H. Villarreal
View a PDF of the paper titled Symbolic Rees algebras, vertex covers and irreducible representations of Rees cones, by Luis A. Dupont and Rafael H. Villarreal
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Abstract: Let G be a simple graph and let J 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 J. 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.
Comments: Algebra Discrete Math., to appear
Subjects: Commutative Algebra (math.AC); Combinatorics (math.CO)
MSC classes: 13F20, 52B20, 05C75
Cite as: arXiv:0712.1249 [math.AC]
  (or arXiv:0712.1249v3 [math.AC] for this version)
  https://doi.org/10.48550/arXiv.0712.1249
arXiv-issued DOI via DataCite
Journal reference: Algebra Discrete Math. 10 (2010), no. 2, 64--86

Submission history

From: Rafael Villarreal H [view email]
[v1] Sat, 8 Dec 2007 00:46:08 UTC (19 KB)
[v2] Fri, 27 Feb 2009 04:11:19 UTC (19 KB)
[v3] Wed, 4 Mar 2009 18:54:01 UTC (19 KB)
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