  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.
