Problem
We consider a network with $m$ nodes and $n$ directed edges. Suppose we can apply labels $y_r \in \mathbb{R}, \ r=1,2,\cdots,m$ to the nodes in the such way that if there is an edge from $r$ to $s$ then $y_r\geq y_s$.
Prove the following: if there is no directed path from node $i$ to $j$, then there exists a labeling of the nodes that satisfies $y_r\geq y_s$(if there is arc from $r$ to $s$) and $y_i<y_j$
What I Have Done
When it comes to network flow problem, I find incident matrix $M$ quite useful and therefore I use $M$ to formulate $y_r\geq y_s$ as $M^Ty \geq 0$. At the same time, there is constraints for network flow problem $Mf=0$ where $f$ stands for flows on different arcs.
From $M^Ty\geq 0$ and $M^Ty=0$, I immediately think of Farkas' lemma, but this is how far I could get since I do not really know to proceed.
Could anyone help me, thank you in advance!