Articulation Points / Cut Vertices

Let G = (V, E) be a connected, undirected graph. An articulation point of G is a vertex whose removal disconnects G. A bridge of G is an edge whose removal disconnects G. A biconnected component of G is a maximal set of edges such that any two edges in the set lie on a common simple cycle. A simple cycle is a cycle with no repeated vertex.

The concept of Articulation Point is explained in the video below. Please excuse my speech impediment while watching the video. Thank you for your patience.

IMPORTANT ANNOUNCEMENT: For the most accurate code please see the code below, rather than looking at the code discussed in the video, as I have later found a bug in the code discussed in the video. Basically the constraints discussed for "parents" apply to all ancestors as well. The corrected version is posted in this chapter, please see the code below the video.


GetArticulationPoints(i, d)
    visited[i] := true
    depth[i] := d
    low[i] := d
    childCount := 0
    isArticulation := false

    for each ni in adj[i] do
        if not visited[ni] then
            parent[ni] := i
            GetArticulationPoints(ni, d + 1)
            childCount := childCount + 1
            if low[ni] ≥ depth[i] then
                isArticulation := true
            low[i] := Min (low[i], low[ni])
        else if ni ≠ parent[i] then
            low[i] := Min (low[i], depth[ni])
    if (parent[i] ≠ null and isArticulation) or (parent[i] = null and childCount > 1) then
        Output i as articulation point

Working Solution:

Java code:

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Python code:

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Time Complexity:

Same as the DFS we are doing to compute Articulation points: O(E + V), where E = total number of edges in the given connected undirected graph, and V = total number of vertices in the given connected undirected graph.

More on Tarjan's Algorithm:


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