Fig 9. Reasons why the bound of Prop. 2 is not tight.

(i) rNNI moves around different arcs may give the same network: the rNNI type-(1) move (u₁u₂, u₁v₁, v₁v₂ → u₁v₂, u₁v₁, v₁u₂) and the rNNI type-(2) move (u₁u₂, u₂v₂, v₁v₂ → u₁v₂, u₂v₂, v₁u₂) on N₁ both give the network $N_1^{\prime }$. (ii) Some of the moves drawn in Fig 8 are not viable rNNI moves: no move of type BR around arc vv₄ of N₂ is allowed, because of the presence of a v₁-v₃ path, which would cause the resulting network to contain a cycle. (iii) The bound of Prop. 2 is tight for some networks: the size of the rNNI neighborhood of network N₃, equal to 12, coincides with the bound.