Bridges and Articulation Points - PowerPoint Presentation

Slide1

Bridges and Articulation Points

Articulation Points

Is a node u of a graph such as if you remove u from the graph then the number of components increases

1 component

3 components

Articulation Points

u is an articulation point if

Right

u is root and has more than 2 children

Wrong

u isn't a root and

there

exists

one of its

subtrees

that has no back edges for an ancestor of u

u

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).

p

Bridges

Is an edge uv of a graph such as if you remove uv from the graph then the number of components increases

1 component

2 components

Bridges

uv is a bridge if and only if it doesn't belong to a cycle.Proof

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

uv is a bridge => doesn't belong to a cycle

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)

uv doesn't belong to a cycle => uv is a bridge

u

v

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

Assume that there is a back edge, then uv is part of a cycle and by the previous slide this is a contradiction

uv is a bridge => there is no back edge...

uv doesn't have a back edge... => uv is a bridge

u

v

u

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.

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.

How to implement?

Let's define for each node vd[v] = time instant that you enter a nodelow[v] = the smallest d that v can reach through its descendants (including v itself)Example:

a

b

c

e

f

g

h

d

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] = 8low[e] = 3low[f] = 2low[g] = 3low[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