From a215f8729ec8a37514a38bdba5921218dcc9b7a6 Mon Sep 17 00:00:00 2001
From: Ashwin <ashwin.kurup@iitgn.ac.in>
Date: Thu, 22 Oct 2026 11:50:58 +0530
Subject: [PATCH] graphs: fix articulation points and add doctests

---
 graphs/articulation_points.py | 116 +++++++++++++++++++++-------------
 1 file changed, 72 insertions(+), 44 deletions(-)

diff --git a/graphs/articulation_points.py b/graphs/articulation_points.py
index 1de20e13752b..1933faa5b731 100644
--- a/graphs/articulation_points.py
+++ b/graphs/articulation_points.py
@@ -1,55 +1,83 @@
 # Finding Articulation Points in Undirected Graph
-def compute_ap(graph) -> None:
+# https://en.wikipedia.org/wiki/Biconnected_component
+
+
+def compute_ap(graph: dict[int, list[int]]) -> list[int]:
+    """
+    Return the articulation points of an undirected graph in increasing order.
+
+    An articulation point is a vertex whose removal disconnects part of the graph.
+    The graph is an adjacency list whose keys are the vertices 0 to n - 1.
+
+    >>> compute_ap({0: [1, 2], 1: [0, 2], 2: [0, 1, 3, 5], 3: [2, 4], 4: [3],
+    ...             5: [2, 6, 8], 6: [5, 7], 7: [6, 8], 8: [5, 7]})
+    [2, 3, 5]
+    >>> compute_ap({0: [5, 4], 1: [3], 2: [5, 3], 3: [1, 2], 4: [5, 0],
+    ...             5: [2, 4, 0]})
+    [2, 3, 5]
+    >>> compute_ap({0: [1], 1: [0, 2], 2: [1]})
+    [1]
+    >>> compute_ap({0: [1, 2], 1: [0, 2], 2: [0, 1]})
+    []
+    >>> compute_ap({0: [1], 1: [0], 2: [3, 4], 3: [2, 4], 4: [2, 3]})
+    []
+    >>> compute_ap({0: [1], 1: [0, 2], 2: [1], 3: [4], 4: [3, 5], 5: [4]})
+    [1, 4]
+    >>> compute_ap({0: []})
+    []
+    >>> compute_ap({})
+    []
+    """
     n = len(graph)
-    out_edge_count = 0
-    low = [0] * n
-    visited = [False] * n
+    discovery = [-1] * n  # order in which each vertex is first reached (-1: not yet)
+    low = [0] * n  # smallest discovery number reachable from the vertex's branch
     is_art = [False] * n
+    timer = 0
 
-    def dfs(root, at, parent, out_edge_count):
-        if parent == root:
-            out_edge_count += 1
-        visited[at] = True
-        low[at] = at
-
+    def dfs(at: int, parent: int) -> int:
+        """Visit ``at`` and return the number of branches the search starts from it."""
+        nonlocal timer
+        discovery[at] = low[at] = timer
+        timer += 1
+        children = 0
         for to in graph[at]:
             if to == parent:
-                pass
-            elif not visited[to]:
-                out_edge_count = dfs(root, to, at, out_edge_count)
+                continue
+            if discovery[to] == -1:
+                children += 1
+                dfs(to, at)
                 low[at] = min(low[at], low[to])
-
-                # AP found via bridge
-                if at < low[to]:
-                    is_art[at] = True
-                # AP found via cycle
-                if at == low[to]:
+                # a branch that cannot get back above ``at`` is cut off without it
+                if parent != -1 and low[to] >= discovery[at]:
                     is_art[at] = True
             else:
-                low[at] = min(low[at], to)
-        return out_edge_count
+                low[at] = min(low[at], discovery[to])
+        return children
 
     for i in range(n):
-        if not visited[i]:
-            out_edge_count = 0
-            out_edge_count = dfs(i, i, -1, out_edge_count)
-            is_art[i] = out_edge_count > 1
-
-    for x in range(len(is_art)):
-        if is_art[x] is True:
-            print(x)
-
-
-# Adjacency list of graph
-graph = {
-    0: [1, 2],
-    1: [0, 2],
-    2: [0, 1, 3, 5],
-    3: [2, 4],
-    4: [3],
-    5: [2, 6, 8],
-    6: [5, 7],
-    7: [6, 8],
-    8: [5, 7],
-}
-compute_ap(graph)
+        if discovery[i] == -1:
+            # the starting vertex is critical only if the search leaves it
+            # in two or more separate directions
+            is_art[i] = dfs(i, -1) > 1
+
+    return [i for i in range(n) if is_art[i]]
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()
+
+    # Adjacency list of graph
+    graph = {
+        0: [1, 2],
+        1: [0, 2],
+        2: [0, 1, 3, 5],
+        3: [2, 4],
+        4: [3],
+        5: [2, 6, 8],
+        6: [5, 7],
+        7: [6, 8],
+        8: [5, 7],
+    }
+    print(compute_ap(graph))