A graph $G=(V,E)$ is said to be vertex-critical if removing a vertex $v\in V$ reduces the chromatic number $\chi(\cdot)$. Edge-criticality is defined in a similar manner. Moreover, $G$ is called contraction-critical if contracting any edge reduces the chromatic number.
1) Are edge- and vertex-criticality equivalent?
2) What is an example of a graph $G$ such that $\omega(G) < \chi(G)$ [where $\omega(G)$ denotes the clique number), and $G$ is vertex-critical, but not contraction-critical?