Removing the vertex also removes the equal endpoint. DFS low-link summaries reveal child subtrees lacking a route above the parent vertex.

Ask whether a child branch can climb above its parent without that parent. Reaching the parent itself is insufficient when the parent vertex disappears.

What the function promises: Return sorted articulation vertices for a simple undirected graph, including disconnected components.

def articulation_points(n, edges):
    graph = [[] for _ in range(n)]
    for u, v in edges:
        graph[u].append(v); graph[v].append(u)
    discovered = [-1] * n
    low = [0] * n
    answer, timer = set(), 0
    def visit(u, parent):
        nonlocal timer
        discovered[u] = low[u] = timer
        timer += 1
        children = 0
        for v in graph[u]:
            if v == parent: continue
            if discovered[v] == -1:
                children += 1
                visit(v, u)
                low[u] = min(low[u], low[v])
                if parent != -1 and low[v] >= discovered[u]:
                    answer.add(u)
            else:
                low[u] = min(low[u], discovered[v])
        if parent == -1 and children > 1:
            answer.add(u)
    for start in range(n):
        if discovered[start] == -1: visit(start, -1)
    return sorted(answer)

assert articulation_points(4, [(0,1),(1,2),(1,3)]) == [1]
assert articulation_points(3, [(0,1),(1,2),(2,0)]) == []

Work and storage: O(V+E) time and O(V) auxiliary storage, including recursion.

When to recognize it: DFS low-link summaries reveal child subtrees lacking a route above the parent vertex.

Check before applying it: Roots and parallel edges need careful treatment; recursion may overflow on deep graphs.

Track edge IDs and skip only the incoming edge, as in the bridge lesson.