Let $G=(\mathcal{V}\_G,\mathcal{A}\_G)$ be an oriented acyclic graph. Assume that $G$ has a unique source $s\in \mathcal{V}\_G$ and a unique sink $t\in \mathcal{V}\_G$. Now, fix $u,v\in \mathcal{V}\_G$ such that $(v,u)\in \mathcal{A}\_G$. Is it true that there exists an oriented path in $G$ of type: $v\_0=s, v\_1, \cdots, v\_i=v, v\_{i+1}=u, \cdots, v\_n=t$? In other words, is there an oriented path containing the arrow $(v,u)$, starting in the source and ending in the sink?



Oriented path in a graph
With the acyclic condition, the answer is yes. Starting at $v$ and repeatedly following any edge exiting the current vertex, you will eventually end up at $t$, by acyclicity and uniqueness of the sink. Thus there exists a path from $v$ to $t$ and, similarly, there exists a path from $s$ to $u$, which shows what you want.


