Fig. 6.

The application of Lemma 6 twice in order to get all reticulations neatly at the top. Left: A network N on a set of taxa $\{a,b\}$ with four reticulations at the top. The network N contains a triangle at $v_{\rho }$ with arcs $(v_{\rho },a_1),(v_{\rho },b_1),$ and $(a_1,b_1)$, where $v_{\rho }$ is the child of the root. Middle: A binary network $N^{\prime }$ obtained by applying the valid rNNI move $(a_2,b_2,b_3)\stackrel{(b_1,b_2)}{\to }(a_2,a_3)$ on N. This illustrates Lemma 6, where the highest 2 horizontal arcs at the top are reoriented. Right: A network $N^{\prime \prime }$ by applying the valid rNNI move $(b_3,a_3,a_4)\stackrel{(b_2,a_3)}{\to }(b_3,b_4)$ to $N^{\prime }$. The highest 3 horizontal arcs at the top are reoriented. The network $N^{\prime \prime }$ has four reticulations neatly at the top