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

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

2007-12-08

arxiv.org/abs/0712.1249v3

Algebra Discrete Math. 10 (2010), no. 2, 64--86

Mathematics

Commutative Algebra

Algebra Discrete Math., to appear

Scientific paper

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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