A connected graph G is 2-edge connected if removal of any edge from G does not disconnect G.  The IsTwoEdgeConnected command returns true if G is 2-edge connected and false otherwise.

TwoEdgeConnectedComponents returns the 2-edge connected components of a graph G.  The output is a list of lists of vertices of G, each being the list of vertices of a component.

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

$\mathrm{IsTwoEdgeConnected}\left(\mathrm{CycleGraph}\left(4\right)\right)$

$\mathrm{true}$

$\mathrm{IsTwoEdgeConnected}\left(\mathrm{PathGraph}\left(4\right)\right)$

$\mathrm{false}$

$G≔\mathrm{Graph}\left(\left\{\left\{a\,b\right\}\,\left\{a\,c\right\}\,\left\{a\,h\right\}\,\left\{a\,i\right\}\,\left\{b\,c\right\}\,\left\{c\,d\right\}\,\left\{d\,e\right\}\,\left\{d\,f\right\}\,\left\{e\,f\right\}\,\left\{h\,i\right\}\right\}\right)\:$

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

$\mathrm{false}$

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

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

$\left[\left[d\,e\,f\right]\,\left[a\,b\,c\,h\,i\right]\right]$