#### Breadth First Search Shortest Path

I
```"""Breath First Search (BFS) can be used when finding the shortest path
from a given source node to a target node in an unweighted graph.
"""

from __future__ import annotations

graph = {
"A": ["B", "C", "E"],
"B": ["A", "D", "E"],
"C": ["A", "F", "G"],
"D": ["B"],
"E": ["A", "B", "D"],
"F": ["C"],
"G": ["C"],
}

class Graph:
def __init__(self, graph: dict[str, list[str]], source_vertex: str) -> None:
"""
Graph is implemented as dictionary of adjacency lists. Also,
Source vertex have to be defined upon initialization.
"""
self.graph = graph
# mapping node to its parent in resulting breadth first tree
self.parent: dict[str, str | None] = {}
self.source_vertex = source_vertex

def breath_first_search(self) -> None:
"""
This function is a helper for running breath first search on this graph.
>>> g = Graph(graph, "G")
>>> g.breath_first_search()
>>> g.parent
{'G': None, 'C': 'G', 'A': 'C', 'F': 'C', 'B': 'A', 'E': 'A', 'D': 'B'}
"""
visited = {self.source_vertex}
self.parent[self.source_vertex] = None
queue = [self.source_vertex]  # first in first out queue

while queue:
vertex = queue.pop(0)
for adjacent_vertex in self.graph[vertex]:
if adjacent_vertex not in visited:

def shortest_path(self, target_vertex: str) -> str:
"""
This shortest path function returns a string, describing the result:
1.) No path is found. The string is a human readable message to indicate this.
2.) The shortest path is found. The string is in the form
`v1(->v2->v3->...->vn)`, where v1 is the source vertex and vn is the target
vertex, if it exists separately.

>>> g = Graph(graph, "G")
>>> g.breath_first_search()

Case 1 - No path is found.
>>> g.shortest_path("Foo")
Traceback (most recent call last):
...
ValueError: No path from vertex: G to vertex: Foo

Case 2 - The path is found.
>>> g.shortest_path("D")
'G->C->A->B->D'
>>> g.shortest_path("G")
'G'
"""
if target_vertex == self.source_vertex:
return self.source_vertex

target_vertex_parent = self.parent.get(target_vertex)
if target_vertex_parent is None:
msg = (
f"No path from vertex: {self.source_vertex} to vertex: {target_vertex}"
)
raise ValueError(msg)

return self.shortest_path(target_vertex_parent) + f"->{target_vertex}"

if __name__ == "__main__":
g = Graph(graph, "G")
g.breath_first_search()
print(g.shortest_path("D"))
print(g.shortest_path("G"))
print(g.shortest_path("Foo"))
```