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EdgeIdeals -- A package for working with the edge ideals of (hyper)graphs

Description

EdgeIdeals is a package to work with the edge ideals of (hyper)graphs.

An edge ideal is a square-free monomial ideal where the generators of the monomial ideal correspond to the edges of the (hyper)graph. An edge ideal complements the Stanley-Reisner correspondence (see SimplicialComplexes) by providing an alternative combinatorial interpretation of the monomial generators.

This package exploits the correspondence between square-free monomial ideals and the combinatorial objects, by using commutative algebra routines to derive information about (hyper)graphs. For some of the mathematical background on this material, see Chapter 6 of the textbook Monomial Algebras by R. Villarreal and the survey paper of T. Ha and A. Van Tuyl ("Resolutions of square-free monomial ideals via facet ideals: a survey," Contemporary Mathematics. 448 (2007) 91-117).

Note: When we use the term "edge ideal of a hypergraph", we are actually referring to the edge ideal of a clutter, a hypergraph where no edge is a subset of another edge. If H is a hypergraph that is not a clutter, then when we form its edge ideal in a similar fashion, some information will be lost because not all of the edges of the hypergraph will correspond to minimal generators, so we require that the edges of hypergraphs do not satisfy any inclusion relations. The edge ideal of a hypergraph is similar to the facet ideal of a simplicial complex, as defined by S. Faridi in "The facet ideal of a simplicial complex," Manuscripta Mathematica 109, 159-174 (2002).

Authors

Version

This documentation describes version 0.1 of EdgeIdeals.

Source code

The source code is in the file EdgeIdeals.m2.

Exports