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
|
The object vertexCovers is a method function.