According to theorem 3.4, a nontrivial graph G has a strong orientation if and only if G is connected and contain no bridges
a)Prove that if $G$ is an nontrivial connected graph with at most 2 bridges, then there exists an orientation $D$ of $G$ having the property that if $u$ and $v$ are any 2 vertices of $D$, there is either a $u-v$ path or a $v-u$ path
b) Show that the statement in a), is false if G contain 3 bridges
for part a) I know that if $G$ contain zero bridge then we are done. So I only need to consider 2 cases
Case 1: $G$ has exactly one bridge
Case 2: $G$ has exactly 2 bridges
For Case 1: Let $D$ be construct by connecting a vertex $v$ to $C_n$ with every arc has same direction. The direction of the edge incident to $v$ doesn't matter, because either way, we still have either a $u-v$ path or a $v-u$ path
For case 2: I use the same graph in case one and add one more vertex called $w$ and connect it to $v$, the edge incident to $w$ must have same direction to the edge connected $v$ and $C_n$, then I will have either a $u-v$ path or a $v-u$ path
for part b) I can't see why it's false if there is 3 bridges, I can repeat case 2, and add another vertex to $w$, can't I?
