Which single node, if it failed, would break the network into disconnected pieces? Those are the ARTICULATION POINTS — the bottlenecks, the single points of failure. One clever depth-first pass finds them all by asking: can any descendant reach back ABOVE me without going through me? If not, I’m critical. It’s how you find fragile routers, key people, and weak bridges. Slide to test-remove each node.
An articulation point (cut vertex) is a vertex whose removal increases the number of connected components — a single point of failure. Tarjan’s DFS finds all of them in one O(V+E) pass using discovery times and LOW-LINK values: low[u] is the earliest vertex reachable from u’s subtree via one back-edge. A non-root u is an articulation point if it has a child v with low[v] ≥ disc[u] (v’s subtree can’t reach above u without u); the DFS root is one iff it has >1 child. The related notion of BRIDGES (critical edges) uses low[v] > disc[u]. It identifies fragile nodes in networks, power grids, and social graphs (structural cut points). A fail-loud self-check throws unless it flags the known cut vertices and not the redundant ones. ◆ real graph algorithms, node-verified.
Articulation points capture SINGLE-vertex fragility; surviving all of them (2-connectivity) doesn’t guarantee robustness to TWO simultaneous failures. The low-link criterion and O(V+E) cost are exact.