A graph G is bi-connected or 2-connected if it is connected and removal of any vertex from G does not disconnect G.  A maximal bi-connected subgraph of G is called a bi-connected component or block of G.

The IsBiconnected command returns true if the input graph is bi-connected. It returns false otherwise.

The Blocks and BiconnectedComponents commands return a list the vertices of all blocks of the graph.  The InducedSubgraph command can be used to extract the blocks of the graph as separate subgraphs.

$\mathrm{with}\left(\mathrm{GraphTheory}\right)\:$

$G≔\mathrm{Graph}\left(\mathrm{Trail}\left(1\,2\,3\,4\,2\right)\,\mathrm{Trail}\left(4\,5\,6\,7\,5\right)\right)$

$G≔\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 7\; vertices\; and\; 8\; edge(s)}$

$\mathrm{DrawGraph}\left(G\right)$

$\mathrm{IsBiconnected}\left(G\right)$

$\mathrm{false}$

$\mathrm{Blocks}\left(G\right)$

$\left[\left[5\,6\,7\right]\,\left[4\,5\right]\,\left[2\,3\,4\right]\,\left[1\,2\right]\right]$

$\mathrm{ArticulationPoints}\left(G\right)$

$\left[2\,4\,5\right]$

$\mathrm{seq}\left(\mathrm{IsBiconnected}\left(\mathrm{InducedSubgraph}\left(G\,B\right)\right)\,B=\mathrm{Blocks}\left(G\right)\right)$

$\mathrm{true},\mathrm{true},\mathrm{true},\mathrm{true}$

Hopcroft and Tarjan, "Efficient algorithms for graph manipulation", CACM. Vol. 16:372-378, 1973. doi:10.1145/362248.362272

"Biconnected component", Wikipedia. http://en.wikipedia.org/wiki/Biconnected_component