NTs have the nice property that they connect stationary points with an index difference of one , compare the example in Section II. If we start at a minimum, we obtain a next saddle point (SP) with index one. A special case is a singular NT that crosses a valley ridge inflection (VRI) point . Normally it connects a minimum with a saddle of index two and two SPs of index one via the VRI. A VRI represents the branching of a valley. With these properties, we can fully investigate the toy PES in Ref. as well as any high-dimensional PES. We can follow a one-dimensional curve by Eq. ( ) in any dimension. Note that the number of steps to reach an SP depends mainly on the step length used, and of course on the length of the path from the minimum to this SP. So we do not need 'drastically different numbers of steps' for different SPs.