HyperGraph -- a class for hypergraphs

Description

This class represents hypergraphs. A hypergraph is a tuple (V,E) of vertices V and edges E which are subsets of the vertices. In this package, all hypergraphs have the additional property that no edge is a subset of any other edge. Hypergraphs of this form are often referred to as clutters.

i1 : R = QQ[w,x,y,z];

i2 : H = hyperGraph(R, {{w,x},{w,y,z},{x,y,z}});

i3 : vertices H

o3 = {w, x, y, z}

o3 : List

i4 : edges H

o4 = {{w, x}, {w, y, z}, {x, y, z}}

o4 : List

i5 : ring H

o5 = R

o5 : PolynomialRing

Hypergraphs are always associated with a polynomial ring whose variables are the vertices of the hypergraph.

Types of hypergraph:

Graph -- a class for graphs

Functions and methods returning a hypergraph:

hyperGraph -- constructor for HyperGraph
hyperGraph(Graph) -- see hyperGraph -- constructor for HyperGraph
hyperGraph(Ideal) -- see hyperGraph -- constructor for HyperGraph
hyperGraph(List) -- see hyperGraph -- constructor for HyperGraph
hyperGraph(MonomialIdeal) -- see hyperGraph -- constructor for HyperGraph
hyperGraph(PolynomialRing,List) -- see hyperGraph -- constructor for HyperGraph

Methods that use a hypergraph:

changeRing(HyperGraph,PolynomialRing,List) -- see changeRing -- replaces vertices with variables of a different ring
chromaticNumber(HyperGraph) -- see chromaticNumber -- computes the chromatic number of a hypergraph
complementGraph(HyperGraph) -- see complementGraph -- returns the complement of a graph or hypergraph
connectedComponents(HyperGraph) -- see connectedComponents -- returns the connected components of a hypergraph
connectedGraphComponents(HyperGraph) -- see connectedGraphComponents -- returns the connected components of a graph
coverIdeal(HyperGraph) -- see coverIdeal -- creates the cover ideal of a (hyper)graph
degreeVertex(HyperGraph,RingElement) -- see degreeVertex -- returns the degree of a vertex
degreeVertex(HyperGraph,ZZ) -- see degreeVertex -- returns the degree of a vertex
deleteEdges(HyperGraph,List) -- see deleteEdges -- returns the (hyper)graph with specified edges removed
edgeIdeal(HyperGraph) -- see edgeIdeal -- creates the edge ideal of a (hyper)graph
edges(HyperGraph) -- see edges -- gets the edges of a (hyper)graph
getEdge(HyperGraph,ZZ) -- see getEdge -- gets the n-th edge in a (hyper)graph
getEdgeIndex(HyperGraph,List) -- see getEdgeIndex -- finds the index of an edge in a HyperGraph
getEdgeIndex(HyperGraph,RingElement) -- see getEdgeIndex -- finds the index of an edge in a HyperGraph
getGoodLeaf(HyperGraph) -- see getGoodLeaf -- returns an edge that is a good leaf
getGoodLeafIndex(HyperGraph) -- see getGoodLeafIndex -- returns the index of an edge that is a good leaf
graph(HyperGraph) -- see graph -- constructor for Graph
hasGoodLeaf(HyperGraph) -- see hasGoodLeaf -- determines if a HyperGraph contains a good leaf
HyperGraph == HyperGraph -- equality
hyperGraphToSimplicialComplex(HyperGraph) -- see hyperGraphToSimplicialComplex -- makes a simplicial complex from a (hyper)graph
incidenceMatrix(HyperGraph) -- see incidenceMatrix -- returns the incidence matrix of a hypergraph
independenceComplex(HyperGraph) -- see independenceComplex -- returns the independence complex of a (hyper)graph
inducedHyperGraph(HyperGraph,List) -- see inducedHyperGraph -- returns the induced subgraph of a (hyper)graph
isCM(HyperGraph) -- see isCM -- determines if a (hyper)graph is Cohen-Macaulay
isConnected(HyperGraph) -- see isConnected -- determines if a (hyper)graph is connected
isConnectedGraph(HyperGraph) -- see isConnectedGraph -- determines if a graph is connected
isEdge(HyperGraph,List) -- see isEdge -- determines if an edge is in a (hyper)graph
isEdge(HyperGraph,RingElement) -- see isEdge -- determines if an edge is in a (hyper)graph
isForest(HyperGraph) -- see isForest -- determines whether a (hyper)graph is a forest
isGoodLeaf(HyperGraph,ZZ) -- see isGoodLeaf -- determines if an edge is a good leaf
isGraph(HyperGraph) -- see isGraph -- determines if a hypergraph is a graph
isLeaf(HyperGraph,RingElement) -- see isLeaf -- determines if an edge (or vertex) is a leaf of a (hyper)graph
isLeaf(HyperGraph,ZZ) -- see isLeaf -- determines if an edge (or vertex) is a leaf of a (hyper)graph
isolatedVertices(HyperGraph) -- see isolatedVertices -- returns all vertices not contained in any edge
isSCM(HyperGraph) -- see isSCM -- determines if a (hyper)graph is sequentially Cohen-Macaulay
lineGraph(HyperGraph) -- see lineGraph -- returns the line graph of a (hyper)graph
neighbors(HyperGraph,List) -- see neighbors -- returns the neighbors of a vertex or list of vertices
neighbors(HyperGraph,RingElement) -- see neighbors -- returns the neighbors of a vertex or list of vertices
neighbors(HyperGraph,ZZ) -- see neighbors -- returns the neighbors of a vertex or list of vertices
numConnectedComponents(HyperGraph) -- see numConnectedComponents -- returns the number of connected components in a (hyper)graph
numConnectedGraphComponents(HyperGraph) -- see numConnectedGraphComponents -- returns the number of connected components in a graph
ring(HyperGraph) -- gives the ring of a (hyper)graph
vertexCoverNumber(HyperGraph) -- see vertexCoverNumber -- find the vertex covering number of a (hyper)graph
vertexCovers(HyperGraph) -- see vertexCovers -- list the minimal vertex covers of a (hyper)graph
vertices(HyperGraph) -- gets the vertices of a (hyper)graph

For the programmer

The object HyperGraph is a type, with ancestor classes HashTable < Thing.

The source of this document is in EdgeIdeals.m2:1455:0.