Symbolic Rees Algebras, Vertex Covers And Irreducible Representations Of Rees Cones

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.