Geodesic
Rees algebras, vertex covers and irreducible representations of Rees cones
Algebra and discrete mathematics, Tome 10 (2010) no. 2, pp. 64-86

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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
Mots-clés : clutter. 
@article{ADM_2010_10_2_a5,
     author = {L. A. Dupont and R. N. Villarreal},
     title = {Rees algebras, vertex covers and irreducible representations of {Rees} cones},
     journal = {Algebra and discrete mathematics},
     pages = {64--86},
     publisher = {mathdoc},
     volume = {10},
     number = {2},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2010_10_2_a5/}
}
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L. A. Dupont; R. N. Villarreal. Rees algebras, vertex covers and irreducible representations of Rees cones. Algebra and discrete mathematics, Tome 10 (2010) no. 2, pp. 64-86. http://geodesic.mathdoc.fr/item/ADM_2010_10_2_a5/