Why is the time complexity of both DFS and BFS O( V + E )

Question

Your sum

v1 + (incident edges) + v2 + (incident edges) + .... + vn + (incident edges)

can be rewritten as

(v1 + v2 + ... + vn) + [(incident_edges v1) + (incident_edges v2) + ... + (incident_edges vn)]

and the first group is O(N) while the other is O(E).

DFS(analysis):

• Setting/getting a vertex/edge label takes O(1) time
• Each vertex is labeled twice
  • once as UNEXPLORED
  • once as VISITED
• Each edge is labeled twice
  • once as UNEXPLORED
  • once as DISCOVERY or BACK
• Method incidentEdges is called once for each vertex
• DFS runs in O(n + m) time provided the graph is represented by the adjacency list structure
• Recall that Σv deg(v) = 2m

BFS(analysis):

• Setting/getting a vertex/edge label takes O(1) time
• Each vertex is labeled twice
  • once as UNEXPLORED
  • once as VISITED
• Each edge is labeled twice
  • once as UNEXPLORED
  • once as DISCOVERY or CROSS
• Each vertex is inserted once into a sequence Li
• Method incidentEdges is called once for each vertex
• BFS runs in O(n + m) time provided the graph is represented by the adjacency list structure
• Recall that Σv deg(v) = 2m

Very simplified without much formality: every edge is considered exactly twice, and every node is processed exactly once, so the complexity has to be a constant multiple of the number of edges as well as the number of vertices.

An intuitive explanation to this is by simply analysing a single loop:

1. visit a vertex -> O(1)
2. a for loop on all the incident edges -> O(e) where e is a number of edges incident on a given vertex v.

So the total time for a single loop is O(1)+O(e). Now sum it for each vertex as each vertex is visited once. This gives

For every V
=> 

    O(1)
    +

    O(e)

=> O(V) + O(E)

Time complexity is O(E+V) instead of O(2E+V) because if the time complexity is n^2+2n+7 then it is written as O(n^2).

Hence, O(2E+V) is written as O(E+V)

because difference between n^2 and n matters but not between n and 2n.