How to create a graph out of biconnected components?

Question

I have a graph and I have identified all of its biconnected components and all of its critical vertexes/articulation points using Tarjan's Algorithm.

I am trying to create a new graph using the biconnected components: the components will be the new vertices and two biconnected components are linked if they share at least one articulation point.

For example, for the graph in the image below, the adjacency lists for the new graph would be:

(1, 3) -> (3, 4, 5), (1, 2)

(1, 2) -> (1, 3)

(3, 4, 5) -> (1, 3)

where (1, 3), (1, 2), (3, 4, 5) are the biconnected components and 1, 3 are articulation points. How could I create the new graph/the adjacency lists in a relatively optimal way? Can I do it while I'm running Tarjan on the graph?

Edit: A biconnected component is a subset of the vertices of the graph where every vertex in this subset will remain connected even if you remove any vertex of the graph. Wiki page for biconnected component

user_4059145 · Answer

Yes you can do it quite easily using DSU(Disjoint Set Union, aka Union Find) data structure.

The key is noticing some observations:

New graph will be a tree or forest of trees (why? - any cycle is bi-connected so it will belong to same component)
The edges of resulting tree will be bridges of original graph (the resulting tree is also known as "Bridge tree")

Now we can already do some tarjan modifications:

for every edge (u,v) if it's a bridge save it for later use
if its not a bridge, use DSU to merge disjoint sets u and v belong to
To construct the resulting tree after tarjan is over we can use saved bridges and remaining disjoint sets,
(opt) we can enumerate disjoint sets if we wish to do so, but because there will never 2 bridges between 2 disjoint sets (that would make a cycle hence contradiction) we can just use disjoint sets identity vertices for new graph

How to create a graph out of biconnected components?

1 Answers