Python breadth first search algorithm
1. Bug

A correct implementation of breadth-first search visits each node at most once, but the code in the post visits some nodes multiple times. In this example, \$C\$ gets visited twice:

&gt;&gt;&gt; graph = {'A': 'BC', 'B': 'C', 'C': 'A'}
&gt;&gt;&gt; bfs(graph, 'A', 'Z')
A
vertex ['B']
vertex ['B', 'C']
B
vertex ['C', 'C']
C
vertex ['C', 'A']
C
vertex ['A']


The problem is that a node is only added to the history list after you pop it from the vertex list. But that means that if there are multiple paths to a node, then it can be added to the vertex list multiple times (as in the example above, where node \$C\$ gets added twice).

Instead, a node should be added to the history list at the same time as you add it to the vertex list.

2. Review


The name queue would be better than vertex (it's a queue of nodes that have been visited in the search but whose neighbours have not yet been considered for visiting).
The queue is implemented using a list. But this is inefficient because popping the leftmost element of a list takes time proportional to the length of the list (because the other items in the list get moved down in the list to fill the space). It is better to use a collections.deque, which has an efficient popleft method.
The name visited would be better than history (it's a collection of nodes that have been visited in the search).
The collection of visited nodes is implemented using a list. But this is inefficient because finding an element in a list takes time proportional to the length of the list (because the element gets compared against each member of the list in turn). It is better to use a set, which has efficient membership determination.
The name neighbour would be better than i (it's a neighbour of node in the graph).
It's pointless to add a node to the queue and then immediately remove it again:

    if i:  #if i is not empty
        vertex.append(i)  
    if i in history:    
       vertex.pop(-1)

Instead of testing for node == end after popping the node from the queue, it would be better to test before adding it to the queue. That way you wouldn't have to wait for the end node to move all the way up the queue.


3. Revised code

from collections import deque

def bfs(graph, start, end):
    """Return True if there is a path from start to end in graph, False if
    not. graph must be a dictionary mapping a node to a collection of its
    neighbours.
    """
    if start == end:
        return True
    queue = deque([start])
    visited = set(queue)
    while queue:
        node = queue.popleft()
        for neighbour in graph.get(node, ()):
            if neighbour not in visited:
                if neighbour == end:
                    return True
                queue.append(neighbour)
                visited.add(neighbour)
    return False