Implement is_articulation to check if vertex is articulation point

CHANGELOG.md

@@ -2,6 +2,10 @@
 
 We follow SemVer as most of the Julia ecosystem. Below you might see the "breaking" label even for minor version bumps -- we use it a bit more loosely to denote things that are not breaking by SemVer's definition but might cause breakage to people using internal or experimental APIs or undocumented implementation details. 
 
+## unreleased
+- `is_articulation(g, v)` for checking whether a single vertex is an articulation point
+
+
 ## v1.14.0 - 2026-02-26
 - **(breaking)** `neighbors`, `inneighbors`, and `outneighbors` now return an immutable `FrozenVector` instead of `Vector`
 - Louvain community detection algorithm

src/Graphs.jl

@@ -430,6 +430,7 @@ export
 
     # biconnectivity and articulation points
     articulation,
+    is_articulation,
     biconnected_components,
     bridges,
 

src/biconnectivity/articulation.jl

@@ -1,8 +1,9 @@
 """
     articulation(g)
 
-Compute the [articulation points](https://en.wikipedia.org/wiki/Biconnected_component)
-of a connected graph `g` and return an array containing all cut vertices.
+Compute the [articulation points](https://en.wikipedia.org/wiki/Biconnected_component) (also
+known as cut or seperating vertices) of an undirected graph `g` and return a vector
+containing all the vertices of `g` that are articulation points.
 
 # Examples
 ```jldoctest
@@ -22,74 +23,136 @@ julia> articulation(path_graph(5))
 function articulation end
 @traitfn function articulation(g::AG::(!IsDirected)) where {T,AG<:AbstractGraph{T}}
     s = Vector{Tuple{T,T,T}}()
-    is_articulation_pt = falses(nv(g))
     low = zeros(T, nv(g))
     pre = zeros(T, nv(g))
 
+    is_articulation_pt = falses(nv(g))
     @inbounds for u in vertices(g)
-        pre[u] != 0 && continue
-        v = u
-        children = 0
-        wi::T = zero(T)
-        w::T = zero(T)
-        cnt::T = one(T)
-        first_time = true
-
-        # TODO the algorithm currently relies on the assumption that
-        # outneighbors(g, v) is indexable. This assumption might not be true
-        # in general, so in case that outneighbors does not produce a vector
-        # we collect these vertices. This might lead to a large number of
-        # allocations, so we should find a way to handle that case differently,
-        # or require inneighbors, outneighbors and neighbors to always
-        # return indexable collections.
-
-        while !isempty(s) || first_time
-            first_time = false
-            if wi < 1
-                pre[v] = cnt
-                cnt += 1
-                low[v] = pre[v]
-                v_neighbors = collect_if_not_vector(outneighbors(g, v))
-                wi = 1
-            else
-                wi, u, v = pop!(s)
-                v_neighbors = collect_if_not_vector(outneighbors(g, v))
-                w = v_neighbors[wi]
-                low[v] = min(low[v], low[w])
-                if low[w] >= pre[v] && u != v
+        articulation_dfs!(is_articulation_pt, g, u, s, low, pre)
+    end
+
+    articulation_points = T[v for (v, b) in enumerate(is_articulation_pt) if b]
+
+    return articulation_points
+end
+
+"""
+    is_articulation(g, v)
+
+Determine whether `v` is an
+[articulation point](https://en.wikipedia.org/wiki/Biconnected_component) of an undirected
+graph `g`, returning `true` if so and `false` otherwise.
+
+See also [`articulation`](@ref).
+
+# Examples
+```jldoctest
+julia> using Graphs
+
+julia> g = path_graph(5)
+{5, 4} undirected simple Int64 graph
+
+julia> articulation(g)
+3-element Vector{Int64}:
+ 2
+ 3
+ 4
+
+julia> is_articulation(g, 2)
+true
+
+julia> is_articulation(g, 1)
+false
+```
+"""
+function is_articulation end
+@traitfn function is_articulation(g::AG::(!IsDirected), v::T) where {T,AG<:AbstractGraph{T}}
+    s = Vector{Tuple{T,T,T}}()
+    low = zeros(T, nv(g))
+    pre = zeros(T, nv(g))
+
+    return articulation_dfs!(nothing, g, v, s, low, pre)
+end
+
+@traitfn function articulation_dfs!(
+    is_articulation_pt::Union{Nothing,BitVector},
+    g::AG::(!IsDirected),
+    u::T,
+    s::Vector{Tuple{T,T,T}},
+    low::Vector{T},
+    pre::Vector{T},
+) where {T,AG<:AbstractGraph{T}}
+    if !isnothing(is_articulation_pt)
+        if pre[u] != 0
+            return is_articulation_pt
+        end
+    end
+
+    v = u
+    children = 0
+    wi::T = zero(T)
+    w::T = zero(T)
+    cnt::T = one(T)
+    first_time = true
+
+    # TODO the algorithm currently relies on the assumption that
+    # outneighbors(g, v) is indexable. This assumption might not be true
+    # in general, so in case that outneighbors does not produce a vector
+    # we collect these vertices. This might lead to a large number of
+    # allocations, so we should find a way to handle that case differently,
+    # or require inneighbors, outneighbors and neighbors to always
+    # return indexable collections.
+
+    while !isempty(s) || first_time
+        first_time = false
+        if wi < 1
+            pre[v] = cnt
+            cnt += 1
+            low[v] = pre[v]
+            v_neighbors = collect_if_not_vector(outneighbors(g, v))
+            wi = 1
+        else
+            wi, u, v = pop!(s)
+            v_neighbors = collect_if_not_vector(outneighbors(g, v))
+            w = v_neighbors[wi]
+            low[v] = min(low[v], low[w])
+            if low[w] >= pre[v] && u != v
+                if isnothing(is_articulation_pt)
+                    if v == u
+                        return true
+                    end
+                else
                     is_articulation_pt[v] = true
                 end
-                wi += 1
             end
-            while wi <= length(v_neighbors)
-                w = v_neighbors[wi]
-                if pre[w] == 0
-                    if u == v
-                        children += 1
-                    end
-                    push!(s, (wi, u, v))
-                    wi = 0
-                    u = v
-                    v = w
-                    break
-                elseif w != u
-                    low[v] = min(low[v], pre[w])
+            wi += 1
+        end
+        while wi <= length(v_neighbors)
+            w = v_neighbors[wi]
+            if pre[w] == 0
+                if u == v
+                    children += 1
                 end
-                wi += 1
+                push!(s, (wi, u, v))
+                wi = 0
+                u = v
+                v = w
+                break
+            elseif w != u
+                low[v] = min(low[v], pre[w])
             end
-            wi < 1 && continue
+            wi += 1
         end
+        wi < 1 && continue
+    end
 
-        if children > 1
+    if children > 1
+        if isnothing(is_articulation_pt)
+            return u == v
+        else
             is_articulation_pt[u] = true
         end
     end
 
-    articulation_points = Vector{T}()
-
-    for u in findall(is_articulation_pt)
-        push!(articulation_points, T(u))
-    end
-
-    return articulation_points
+    return isnothing(is_articulation_pt) ? false : is_articulation_pt
 end

test/biconnectivity/articulation.jl

@@ -22,18 +22,30 @@
         art = @inferred(articulation(g))
         ans = [1, 7, 8, 12]
         @test art == ans
+        @test art == findall(is_articulation.(Ref(g), vertices(g)))
     end
     for level in 1:6
         btree = Graphs.binary_tree(level)
         for tree in test_generic_graphs(btree; eltypes=[Int, UInt8, Int16])
             artpts = @inferred(articulation(tree))
             @test artpts == collect(1:(2 ^ (level - 1) - 1))
+            @test artpts == findall(is_articulation.(Ref(tree), vertices(tree)))
         end
     end
 
     hint = blockdiag(wheel_graph(5), wheel_graph(5))
     add_edge!(hint, 5, 6)
     for h in test_generic_graphs(hint; eltypes=[Int, UInt8, Int16])
-        @test @inferred(articulation(h)) == [5, 6]
+        art = @inferred(articulation(h))
+        @test art == [5, 6]
+        @test art == findall(is_articulation.(Ref(h), vertices(h)))
     end
+
+    # graph with disconnected components
+    g = path_graph(5)
+    es = collect(edges(g))
+    g2 = Graph(vcat(es, [Edge(e.src + nv(g), e.dst + nv(g)) for e in es]))
+    @test articulation(g) == [2, 3, 4] # a single connected component
+    @test articulation(g2) == [2, 3, 4, 7, 8, 9] # two identical connected components
+    @test articulation(g2) == findall(is_articulation.(Ref(g2), vertices(g2)))
 end