Let $G$ be a directed weighted graph with the cost $c: E(G) \rightarrow \mathbb{R}$ and $s, t \in V(G)$. Find a path $P$ that connects $s, t$ such that $max\{c(e) | e \in E(P)\}$ is minimized.
I know that in an undirected graph, it is sufficient to find the optimal spanning tree (the spanning tree with minimal weight) and then for any given $s, t$ the way in the tree will minimize the cost of the most expensive edge.
Is there a way I can use this knowledge in the case of a directed graph?