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arXiv · 0712.1249

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

Luis A. Dupont·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.

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Luis A. Dupont, Rafael H. Villarreal. 2009-03-04. Symbolic Rees algebras, vertex covers and irreducible representations of Rees cones. https://arxiv.org/abs/0712.1249

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