Bicomponents

Bicomponents[g]
gives the biconnected components of the undirected graph g.

A biconnected component is a maximal subgraph that has no cutpoint, where a cutpoint is a vertex v such that the subgraph becomes disconnected if v and all its edges are removed.

This shows that a simple line with two vertices is biconnected:

This defines a small graph:

The graph has four bicomponents, one for each cycle and two for the line joining the cycles:

Needs["GraphUtilities`"]

This shows that a simple line with two vertices is biconnected:

Out[3]=

Out[4]=

Needs["GraphUtilities`"]

This defines a small graph:

Out[3]=

The graph has four bicomponents, one for each cycle and two for the line joining the cycles:

Out[4]=

This defines a graph:

This shows its biconnected components and connected components:

The result from WeakComponents is always smaller than that of