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 $G$.