Variant of greedy algorithm for vertex cover


Does the following approximation algorithm for vertex cover also have an approximation ratio of 2? Why? Why not?

Input: $ G = (V,E)$

  1. Set $ C \gets \emptyset$ and $ E’ \gets E$ .
  2. while $ E’ \neq \emptyset$ do:
    • Pick any edge $ (u,v) \in E’$ .
    • $ C \gets C \cup \{u\}$ .
    • Remove from $ E’$ all edges incident to $ u$ .
  3. return $ C$ .