graphs – Does every DAG have at most one “universal source”?

Suppose towards a contradiction that a DAG $G=(V,E)$ on $n$ vertices has more than one universal source and let $u,v in V$ be two distinct universal sources of $G$.

Since $mbox{out-deg}(u)=n-1$, and $G$ contains no self-loops, vertex $u$ has edges towards every other vertex in $G$. In particular $(u,v) in E$. As a consequence $mbox{in-deg}(v) ge 1 > 0$ showing that $v$ cannot be a universal source, yielding the sought contradiction.