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coverIdeal -- creates the cover ideal of a (hyper)graph



Returns the monomial ideal generated by the minimal vertex covers. This is also the Alexander dual of the edge ideal of the hypergraph H.

i1 : S = QQ[a,b,c,d,e,f];
i2 : k6 = completeGraph S  -- complete graph on 6 vertices

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

o2 : Graph
i3 : coverIdeal k6 -- each generator corresponds to a minimal vertex of k6

o3 = monomialIdeal (a*b*c*d*e, a*b*c*d*f, a*b*c*e*f, a*b*d*e*f, a*c*d*e*f,

o3 : MonomialIdeal of S
i4 : h = hyperGraph {a*b*c,c*d,d*e*f}

o4 = HyperGraph{"edges" => {{a, b, c}, {c, d}, {d, e, f}}}
                "ring" => S
                "vertices" => {a, b, c, d, e, f}

o4 : HyperGraph
i5 : coverIdeal h

o5 = monomialIdeal (a*d, b*d, c*d, c*e, c*f)

o5 : MonomialIdeal of S
i6 : dual coverIdeal h == edgeIdeal h

o6 = true

See also

Ways to use coverIdeal :

For the programmer

The object coverIdeal is a method function.