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: We require all hypergraphs to be clutters, which are hypergraphs in which no edge is a subset of another. If $H$ is a hypergraph that is not a clutter, then the edge ideal of $H$ is indistinguishable from the edge ideal of the clutter of minimal edges in $H$. (Edges of $H$ that are supersets of other edges would not appear as minimal generators of the edge ideal of $H$.) 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).
Version 1.0.0 of this package was accepted for publication in volume 1 of The Journal of Software for Algebra and Geometry: Macaulay2 on 2009-06-27, in the article EdgeIdeals: a package for (hyper)graphs. That version can be obtained from the journal or from the Macaulay2 source code repository.
This documentation describes version 1.0.2 of EdgeIdeals.
The source code from which this documentation is derived is in the file EdgeIdeals.m2.