1. Bridges and Articulation Points

2. Articulation Points
Is a node u of a graph such as if you remove u from the graph then the number of components increases
3 components
1 component

3. Articulation Points u is an articulation point if
u is root and has more than 2 children
u isn't a root and there exists one of its subtrees that has no back edges for an ancestor of u p u
Right
Wrong
The second and third subtree all have back edges to p (an ancestor of u), while the first one hasn't, therefore u is an articulation point (notice that if you remove it, the first subtree will be disconnected from the rest of the graph).

4. Bridges
Is an edge uv of a graph such as if you remove uv from the graph then the number of components increases
2 components
1 component

5. Bridges uv is a bridge if and only if it doesn't belong to a cycle.
Proof
uv is a bridge => doesn't belong to a cycle
uv doesn't belong to a cycle => uv is a bridge u v u v
uv is not in a cycle so if uv is removed there is no other way to get from u to v, so you are cutting the graph into 2 components (one with u and another with v)
If you remove uv there is another path that takes from u to v (the graph is still connected), so uv is not a bridge

6. Bridges The definition on the previous slide gives us that:
uv is a bridge ⇔ there is no back edge from a descendant of u to an ancestor of u
uv is a bridge => there is no back edge...
uv doesn't have a back edge... => uv is a bridge u u v v
Assume that uv is not a bridge, then once it’s removed it must have a path between u and v, but we are assuming that there is no back edge, contradiction.
Assume that there is a back edge, then uv is part of a cycle and by the previous slide this is a contradiction

7. Interesting fact
If you remove a bridge of a graph the number of components increases by exactly one but if you remove an articulation point the number of components can increase by more than one.

8. How to implement? Let's define for each node v Example:
d[v] = time instant that you enter a node
low[v] = the smallest d that v can reach through its descendants (including v itself)
Example:
d[a] = 1
d[b] = 2
d[c] = 3
d[d] = 8
d[e] = 4
d[f] = 7
d[g] = 5
d[h] = 6
low[a] = 1
low[b] = 1
low[c] = 1
low[d] = 8
low[e] = 3
low[f] = 2
low[g] = 3
low[h] = 6
u is an articulation point if
low[v] >= d[u] for some children v
uv is a bridge if
low[v] > d[u] for some children v
Notice that low[e] = 3 and not 1 a b c d e f g h