Articulation Points

A network can be connected yet depend on one vulnerable vertex. Remove that vertex and every edge touching it: if the number of connected components increases, the vertex is an articulation point, also called a cut vertex. The question concerns vertex failure, rather than which path is shortest or which edges form the cheapest spanning tree.

Experiment: remove a vertex, then explain the split

Open the Articulation Points explorer. All inputs below are supported native controls. The complete graph analysis is immediate; the return replay separately reveals its DFS certificate.

1. Choose Joined cycles under Graph example. With Inspect vertex set to 2, predict the two remaining groups. Read one component before, two after, increase one and selected cut Yes.
2. Change Graph view to Vertex removed. The selected vertex is a dashed ghost, its incident edges disappear, and the remaining colors distinguish {0, 1} from {3, 4, 5}. The ghost is excluded from all counts.
3. Return to Original graph. Set DFS return step to 2, or press Next return twice. The active vertex is 4, replay low[4] and low[5] are 3, and only 3 has been confirmed so far. Other replay low entries remain Pending.
4. Press Finish returns. Confirm final low values 0, 0, 0, 3, 3, 3 and confirmed cuts 2, 3. Previous return reverses one reveal; Rewind returns clears the replay. Play returns advances automatically and Pause returns stops it.
5. Choose Add a bypass. One extra edge, 1–4, removes both cut vertices. Inspect vertex 2 again: the graph remains connected. Compare the two removal pictures rather than inferring reliability from degree alone.
6. Choose Star root and inspect 0. The increase is two because one component becomes three. Finish the replay: root 0 is confirmed only at its own return, after all three children have returned. Inspect leaf 1 instead; that leaf is not a cut vertex.
7. Choose Disconnected graph. Inspect 1: three original components become four, with only vertex 1 a cut vertex. The isolated vertex 6 is counted both before and after this removal. Inspect 6 instead: deleting the isolate reduces the total from three to two and does not make it an articulation point.
8. Edit Undirected edges to include a self-loop such as 1 1. An error replaces the graph and old results. Reset experiment restores the original seven edges and query. Use Inspect vertex or click a node to select it. Pan the background, use arrow keys or Home, and compare Reset view: navigation changes no graph result. Narrow tables scroll horizontally.