This is an option for the function monodromySolve which uses a complete graph. For instance, the user can choose to have 4 vertices and 2 edges connecting each pair of vertices. Then the homotopy will run on the complete graph on 4 vertices, where each edge is doubled.

i1 : R = CC[a,b,c,d][x,y]; |

i2 : polys = polySystem {a*x+b*y^2,c*x*y+d}; |

i3 : monodromySolve(polys,GraphInitFunction => completeGraphInit) o3 = (HomotopyNode{...7...}, 4) o3 : Sequence |

The object completeGraphInit is a function closure.