Fig. 6

A tree \(\Gamma \) is depicted in gray and some arcs of N are depicted in black. Recall that t is the number of children of x and \(Z_i:=\bigcup _{1\le j\le i}\Gamma _{v_j}\). Note that \(x\in \hbox {Succ}_{N}^{\uparrow }{(Z_2)}\setminus \hbox {Succ}_{N}^{\uparrow }{(\Gamma _x)}\) since x is an ancestor of a node of \(\Gamma _{v_2}\) in N. Note that x is a reticulation of N with parents y (drawn) and z (not drawn) with \(y<_\Gamma v_2<_\Gamma x<_\Gamma z\). Thus, \(z\in \hbox {Pred}_{N}^{\downarrow }{(x)}\) but \(y\in \hbox {Pred}_{N}^{\uparrow v_2}{(x)}\subseteq \hbox {Pred}_{N}^{\uparrow }{(x)}\). Finally, note that \(\hbox {YW}_{x}^{\Gamma }=\hbox {Pred}_{N}^{\downarrow }{(\Gamma _x)}\cup \hbox {Succ}_{N}^{\uparrow }{(\Gamma _x)}\) and \(\bigcup _{i\le t}\hbox {YW}_{{v_i}}^{\Gamma }\subseteq \hbox {YW}_{x}^{\Gamma }\uplus \{x\}\)