The curve complex has dead ends

It is proved that the curve graph $C^1(\Sigma)$ of a surface $\Sigma_{g,n}$ has a local pathology that had not been identified as such: there are vertices $\alpha,\beta$ in $C^1(\Sigma)$ such that $\beta$ is a dead end of every geodesic joining $\alpha$ to $\beta$. It also has double dead-ends. Every dead end has depth 1.