Explanation of runtimes of BFS and DFS

Question

Why are the running times of BFS and DFS O(V+E), especially when there is a node that has a directed edge to a node that can be reached from the vertex, like in this example in

Answer 1

You visit every edge at most twice. There are E edges. So there will be 2*E edge visit operations. Plus the nodes those have no edges or in other words, with degree 0. There can be at most V such nodes. So the complexity turns out to be, O(2*E + V) = O(E + V)