Combinatorics of symbolic Rees algebras of edge ideals of clutters

TL;DR

This paper provides a combinatorial framework for understanding the minimal generators of symbolic Rees algebras of edge ideals of clutters, linking them to irreducible parallelizations and enabling computation via Hilbert bases.

Contribution

It introduces a combinatorial description of minimal generators of symbolic Rees algebras of edge ideals, connecting them to irreducible parallelizations and offering computational methods.

Findings

Minimal generators correspond to irreducible parallelizations

Results on symbolic Rees algebras of perfect graphs and clutters

Method to compute irreducible parallelizations using Hilbert bases

Abstract

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.