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), see vertices -- gets the vertices of a (hyper)graph

For the programmer

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