p

```
"""
An edge is a bridge if, after removing it count of connected components in graph will
be increased by one. Bridges represent vulnerabilities in a connected network and are
useful for designing reliable networks. For example, in a wired computer network, an
articulation point indicates the critical computers and a bridge indicates the critical
wires or connections.
For more details, refer this article:
https://www.geeksforgeeks.org/bridge-in-a-graph/
"""
def __get_demo_graph(index):
return [
{
0: [1, 2],
1: [0, 2],
2: [0, 1, 3, 5],
3: [2, 4],
4: [3],
5: [2, 6, 8],
6: [5, 7],
7: [6, 8],
8: [5, 7],
},
{
0: [6],
1: [9],
2: [4, 5],
3: [4],
4: [2, 3],
5: [2],
6: [0, 7],
7: [6],
8: [],
9: [1],
},
{
0: [4],
1: [6],
2: [],
3: [5, 6, 7],
4: [0, 6],
5: [3, 8, 9],
6: [1, 3, 4, 7],
7: [3, 6, 8, 9],
8: [5, 7],
9: [5, 7],
},
{
0: [1, 3],
1: [0, 2, 4],
2: [1, 3, 4],
3: [0, 2, 4],
4: [1, 2, 3],
},
][index]
def compute_bridges(graph: dict[int, list[int]]) -> list[tuple[int, int]]:
"""
Return the list of undirected graph bridges [(a1, b1), ..., (ak, bk)]; ai <= bi
>>> compute_bridges(__get_demo_graph(0))
[(3, 4), (2, 3), (2, 5)]
>>> compute_bridges(__get_demo_graph(1))
[(6, 7), (0, 6), (1, 9), (3, 4), (2, 4), (2, 5)]
>>> compute_bridges(__get_demo_graph(2))
[(1, 6), (4, 6), (0, 4)]
>>> compute_bridges(__get_demo_graph(3))
[]
>>> compute_bridges({})
[]
"""
id_ = 0
n = len(graph) # No of vertices in graph
low = [0] * n
visited = [False] * n
def dfs(at, parent, bridges, id_):
visited[at] = True
low[at] = id_
id_ += 1
for to in graph[at]:
if to == parent:
pass
elif not visited[to]:
dfs(to, at, bridges, id_)
low[at] = min(low[at], low[to])
if id_ <= low[to]:
bridges.append((at, to) if at < to else (to, at))
else:
# This edge is a back edge and cannot be a bridge
low[at] = min(low[at], low[to])
bridges: list[tuple[int, int]] = []
for i in range(n):
if not visited[i]:
dfs(i, -1, bridges, id_)
return bridges
if __name__ == "__main__":
import doctest
doctest.testmod()
```