Combinatorics of Symbolic Rees Algebras of Edge Ideals of Clutters

Let C be a clutter and let I be its edge ideal. We present a combinatorial description of the minimal generators of the symbolic Rees algebra Rs(I) of I . It is shown that the minimal generators of Rs(I) are in one to one correspondence with the irreducible parallelizations of C. From our description some major results on symbolic Rees algebras of perfect graphs and clutters will follow. As a byproduct, we give a method, using Hilbert bases, to compute all irreducible parallelizations of C along with all the corresponding vertex covering numbers. In particular, we can decide whether any given clutter is irreducible and compute all irreducible induced subclutters of C. If C is a graph, we obtain all odd holes and antiholes of C.