This function returns the vertex covering number of a (hyper)graph. The vertex covering number is the size of smallest vertex cover of the (hyper)graph. This corresponds to the smallest degree of a generator of the cover ideal of the (hyper)graph.
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 : vertexCoverNumber g o3 = 2 |
i4 : S = QQ[a..e]; |
i5 : g = graph {a*b,a*c,a*d,a*e,b*c,b*d,b*e,c*d,c*e,d*e} -- the complete graph K_5
o5 = Graph{edges => {{a, b}, {a, c}, {b, c}, {a, d}, {b, d}, {c, d}, {a, e}, {b, e}, {c, e}, {d, e}}}
ring => S
vertices => {a, b, c, d, e}
o5 : Graph
|
i6 : vertexCoverNumber g o6 = 4 |
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 : vertexCoverNumber(h) o8 = 2 |
The object vertexCoverNumber is a method function.