I'm trying to understand the next proof
I don't understand the way the author chooses the set $S$, I mean, which is the criterion? I've been thinking in two possibilities:
- The only possible arcs from $S$ to V(T)-S are $<x,w>,<x,z>$ or $<x,v>$, but this contadicts what he says next that $S\cup \{w\}$ contradicts the maximality of $S$.
2.The only possible arcs from $S$ to $V(T)-S$ have their head on $z,w$ or $v$, of course if the belong to $V(T)-S$. About this one, it is true that $S\cup \{w\}$ contradicts the maximality of $S$ but the existence of $w_1\in V(T)-S$ which dominates $w$, not necessarily fixes that, unless there is another vertex in $V(T)-S$ dominated by $w$, more over, in this way $S\cup \{w_1\}$ also contradicts the maximality of $S$ unless ther exists another vertex in $V(T)-S$ dominated by $w_1$.
Perhaps I'm missunderstanding the idea about $S$, so I need your help here, I see something weird in this proof. Thanks in advance.