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# vertexCoverNumber -- find the vertex covering number of a (hyper)graph

## Synopsis

• Usage:
c = vertexCoverNumber(H)
• Inputs:
• H, , the input
• Outputs:
• c, an integer, the vertex covering number

## Description

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