A vertex v in a graph G is an articulation point of G if removing it and its incident edges increases the number of connected components of G.

ArticulationPoints(G) returns a list of the vertices of G which are articulation points.

The articulation points of a graph can be computed when traversing the graph using depth-first-search in linear time in the size of the graph.

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

$\mathrm{P5}≔\mathrm{PathGraph}\left(5\right)$

$\mathrm{P5}≔\mathrm{Graph\; 1:\; an\; undirected\; unweighted\; graph\; with\; 5\; vertices\; and\; 4\; edge(s)}$

$\mathrm{ArticulationPoints}\left(\mathrm{P5}\right)$

$\left[2\,3\,4\right]$

$\mathrm{numelems}\left(\mathrm{ConnectedComponents}\left(\mathrm{P5}\right)\right)$

$1$

$G≔\mathrm{DeleteVertex}\left(\mathrm{P5}\,3\right)$

$G≔\mathrm{Graph\; 2:\; an\; undirected\; unweighted\; graph\; with\; 4\; vertices\; and\; 2\; edge(s)}$

$\mathrm{numelems}\left(\mathrm{ConnectedComponents}\left(G\right)\right)$

$2$

$\mathrm{C5}≔\mathrm{CycleGraph}\left(5\right)$

$\mathrm{C5}≔\mathrm{Graph\; 3:\; an\; undirected\; unweighted\; graph\; with\; 5\; vertices\; and\; 5\; edge(s)}$

$\mathrm{ArticulationPoints}\left(\mathrm{C5}\right)$