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vertexCovers -- list the minimal vertex covers of a (hyper)graph

Synopsis

Description

This function returns the minimal vertex covers of a (hyper)graph. A vertex cover is a subset of the vertices such that every edge of the (hyper)graph has non-empty intersection with this set. The minimal vertex covers are given by the minimal generators of the cover ideal of H.

i1 : S = QQ[a..d];
 
i2 : g = graph {a*b,b*c,c*d,d*a} -- the four cycle

o2 = Graph{"edges" => {{a, b}, {b, c}, {a, d}, {c, d}}}
           "ring" => S
           "vertices" => {a, b, c, d}

o2 : Graph
 
i3 : vertexCovers g

o3 = {a*c, b*d}

o3 : List
 
i4 : coverIdeal g

o4 = monomialIdeal (a*c, b*d)

o4 : MonomialIdeal of S
 
i5 : flatten entries gens coverIdeal g == vertexCovers g

o5 = true
 
i6 : S = QQ[a..e];
 
i7 : h = hyperGraph {a*b*c,a*d,c*e,b*d*e}

o7 = HyperGraph{"edges" => {{a, b, c}, {a, d}, {c, e}, {b, d, e}}}
                "ring" => S
                "vertices" => {a, b, c, d, e}

o7 : HyperGraph
 
i8 : vertexCovers(h)

o8 = {a*b*c, c*d, a*e, b*d*e}

o8 : List

See also

Ways to use vertexCovers:

For the programmer

The object vertexCovers is a method function.