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 Algebra Discrete Math., 2010, Volume 10, Issue 2, Pages 64–86 (Mi adm49)

RESEARCH ARTICLE

Rees algebras, vertex covers and irreducible representations of Rees cones

L. A. Dupont, R. N. Villarreal

Departamento de Matem'aticas,Centro de Investigacon y de Estudios, Avanzados del IPN, Apartado Postal 14–740, 07000 Mexico City, D.F.

Abstract: 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 $G$.

Keywords: edge ideal, symbolic Rees algebras, perfect graph, irreducible vertex covers, irreducible graph, Alexander dual, blocker, clutter.

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Citation: L. A. Dupont, R. N. Villarreal, “Rees algebras, vertex covers and irreducible representations of Rees cones”, Algebra Discrete Math., 10:2 (2010), 64–86

Citation in format AMSBIB
\Bibitem{DupVil10} \by L.~A.~Dupont, R.~N.~Villarreal \paper Rees algebras, vertex covers and irreducible representations of Rees cones \jour Algebra Discrete Math. \yr 2010 \vol 10 \issue 2 \pages 64--86 \mathnet{http://mi.mathnet.ru/adm49} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2884744} \zmath{https://zbmath.org/?q=an:1224.13022} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Martinez-Bernal J., Renteria-Marquez C., Villarreal R.H., “Combinatorics of Symbolic Rees Algebras of Edge Ideals of Clutters”, Commutative Algebra and its Connections to Geometry, Contemporary Mathematics, 555, eds. Corso A., Polini C., 2011, 151–164
2. Martinez-Bernal J., Morey S., Villarreal R.H., “Associated Primes of Powers of Edge Ideals”, Collect. Math., 63:3 (2012), 361–374
3. Bayati Sh., Rahmati F., “Squarefree Vertex Cover Algebras”, Commun. Algebr., 42:4 (2014), 1518–1538
4. Tran T., Ziegler G.M., “Extremal Edge Polytopes”, Electron. J. Comb., 21:2 (2014), P2.57
5. Constantinescu A., Pournaki M.R., Fakhari S.A.S., Terai N., Yassemi S., “Cohen-Macaulayness and Limit Behavior of Depth For Powers of Cover Ideals”, Commun. Algebr., 43:1, SI (2015), 143–157
6. Flores-Mendez A., Gitler I., Reyes E., “Implosive Graphs: Square-Free Monomials on Symbolic Rees Algebras”, J. Algebra. Appl., 16:8 (2017), 1750145
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