I'm studying for a graph theory exam and am stumped on one of the practice questions:
Suppose $G$ is $2$-vertex-connected. Show that for any distinct vertices $x$, $y$, $z$ of $G$ there exists a path from $x$ to $y$ containing $z$.
I've tried working with ear decompositions and Menger's theorem (since those are the two major focuses we had in class about 2-connected graphs) but I keep ending up with a walk from $x$ to $y$ containing $z$, which isn't good enough because there's no guarantee that $z$ will exist in the resulting path.