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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.
  • Bicomponents treats the input g as an undirected graph.
Needs["GraphUtilities`"]
This shows that a simple line with two vertices is biconnected:
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Needs["GraphUtilities`"]
This defines a small graph:
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The graph has four bicomponents, one for each cycle and two for the line joining the cycles:
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