Vertices on Quasigeodesics

Theorem 16.5 demonstrated the importance in our context of the number of vertices on a quasigeodesic.quasigeodesicnumber of vertices If, as we conjecture in Open Problem 18.15 , every convex polyhedron P has a quasigeodesic \(\mathcal {Q}\) containing at most one vertex, then the vertex-mergingvertex-merging described in that theorem leads to an unfoldingunfoldingonto cylinder of P to a cylinder \(\mathcal {L}\) and then to a non-overlapping unfolding, an anycut-netnetanycut- for P.