[Paper Review] Symbolic Rees algebras, vertex covers and irreducible representations of Rees cones

Luis A. Dupont, Rafael H. Villarreal|arXiv (Cornell University)|Dec 8, 2007

Commutative Algebra and Its Applications24 references21 citations

TL;DR

This paper provides a graph-theoretic characterization of minimal generators of symbolic Rees algebras for edge ideals and their Alexander duals using irreducible vertex covers and Hilbert bases of Rees cones. It establishes that irreducible binary b-vertex covers of the blocker of a graph correspond to irreducible induced subgraphs, enabling a method to compute all odd holes and anti-holes via Hilbert bases, and constructs high-degree minimal generators for symbolic Rees algebras of cone constructions on graphs.

ABSTRACT

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.

Motivation & Objective

• To describe the minimal generators of the symbolic Rees algebra of the edge ideal of a graph using graph-theoretic invariants.
• To characterize irreducible b-vertex covers of the blocker of a graph in terms of irreducible induced subgraphs.
• To develop a method using Hilbert bases of Rees cones to compute all irreducible induced subgraphs, including odd holes and anti-holes.
• To construct irreducible b-vertex covers of high degree relative to the number of vertices in the blocker of a graph.
• To establish a connection between symbolic Rees algebra generators and combinatorial structures in clutters via polyhedral geometry.

Proposed method

• Use of the Rees cone of the edge ideal to represent the symbolic Rees algebra as a rational polyhedral cone in R^{n+1}.
• Application of integral Hilbert bases to identify minimal generators of the symbolic Rees algebra via irreducible representations of the Rees cone.
• Definition of b-vertex covers and irreducible b-vertex covers as vectors in Z^{n+1} satisfying specific linear inequalities derived from minimal vertex covers of the clutter.
• Construction of the blocker of a graph G as the clutter of minimal vertex covers of G, and study of its symbolic Rees algebra via irreducible covers.
• Use of the cone construction on a graph G to generate higher-dimensional clutters G_r, and derivation of degree bounds for minimal generators of R_s(I(G_r)).
• Leveraging the fact that a vector (a_1, ..., a_n, -b) in Z^{n+1} corresponds to a minimal generator of the symbolic Rees algebra if and only if it defines a facet of the Rees cone via an irreducible b-cover.

Experimental results

Research questions

• RQ1 How can the minimal generators of the symbolic Rees algebra of the edge ideal of a graph be described using graph-theoretic invariants?
• RQ2 What is the relationship between irreducible b-vertex covers of the blocker of a graph and its irreducible induced subgraphs?
• RQ3 Can Hilbert bases of the Rees cone be used to systematically compute all irreducible induced subgraphs, including odd holes and anti-holes?
• RQ4 How can one construct irreducible b-vertex covers of the blocker with high degree relative to the number of vertices?
• RQ5 What is the structure of the symbolic Rees algebra of the edge ideal of a cone over a graph, and what are the degrees of its minimal generators?

Key findings

• Irreducible binary b-vertex covers of the blocker of a graph correspond exactly to irreducible induced subgraphs of the original graph.
• The method using Hilbert bases of the Rees cone allows for the systematic computation of all irreducible induced subgraphs, including all odd holes and anti-holes.
• For a graph G_r obtained by r-fold cone construction on a graph G with n vertices and vertex covering number g, the symbolic Rees algebra R_s(I(G_r)) has a minimal generator of degree n + (r-1)(n - g) in the variable t.
• In the case of an odd cycle C_s of length s = 2k+1, the symbolic Rees algebra of its r-fold cone has a minimal generator x_1⋯x_s x_{s+1}^k⋯x_{s+r}^k t^{rk + k + 1}, showing that the t-degree can exceed the number of vertices.
• The construction of the vector β = (a_1, ..., a_n, (∑a_i - b), -∑a_i) with a_i ≥ 1 yields a facet of the Rees cone of the blocker, proving that the corresponding monomial is a minimal generator of the symbolic Rees algebra.
• The existence of linearly independent characteristic vectors of minimal vertex covers ensures that the constructed facet has dimension n+1, confirming the minimality of the generator.