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

The relation between facets of Rees cones of clutters and irreducible b-vertex covers is examined. Let G be a simple graph and let Ic(G) be its ideal of vertex covers. We give a graph theoretical description of the irreducible bvertex covers of G, i.e., we describe the minimal generators of the symbolic Rees algebra of Ic(G). As an application we recover an explicit description of the edge cone of a graph. 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 are in one to one correspondence with the irreducible induced subgraphs of G. As a byproduct we obtain a method, using Hilbert bases, to obtain all irreducible induced subgraphs ofG. Irreducible graphs are studied. We show how to build 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.