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

TL;DR

This paper explores the algebraic and combinatorial structures of vertex covers and Rees algebras in graphs, providing new graph-theoretic descriptions and methods to identify irreducible subgraphs, including odd holes and antiholes.

Contribution

It offers a novel graph-theoretic characterization of irreducible b-vertex covers and their relation to Rees algebras, advancing understanding of algebraic invariants in graph theory.

Findings

Describes minimal generators of symbolic Rees algebra of vertex cover ideal

Characterizes irreducible induced subgraphs via irreducible binary b-vertex covers

Provides a method to find all irreducible induced subgraphs, including odd holes and antiholes

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.