# Shortest path

Let $L(P) = \sum_{e \in P} \ell(e)$ and $C(P) = \max_{e \in P} \ell(e)$.

$L(1, 2, 3, 4, 5) \mapsto 1+2+3+4+5 = 15$

$C(1, 2, 3, 4, 5) \mapsto \max(1,2,3,4,5) = 5$

You should provide a graph $G$ and vertices $s$ and $t$ and path $P$ from $s$ to $t$ such that:

$$C(P) = \min_{P' \text{ from $s$ to $t$}} C(P') $$

and
$$L(P) \ne \min_{P' \text{ from $s$ to $t$}} L(P') $$

I.e. for the last equation you should provide a path $P'$ from $s$ to $t$ such that $L(P') < L(P)$.