Biconnected Components
(graph/biconnected_components.hpp)

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Last update: 2026-08-13 01:41:40+09:00
Include: #include "graph/biconnected_components.hpp"

Overview

This header decomposes an undirected graph into maximal vertex-biconnected blocks. Every active edge belongs to exactly one block. An articulation vertex belongs to two or more blocks, while an isolated vertex forms a singleton block with no edges.

Pass the result to block_cut_tree from graph/block_cut_tree.hpp to build the block-cut forest and original-vertex-to-node mappings.

Graph Requirements

Build the graph with Graph<T>::add_edge. Directed edges and self-loops are not supported. Parallel edges are supported because traversal identifies edges by their shared edge IDs. Inactive edges are ignored; a vertex incident only to inactive edges is treated as isolated.

The traversal is iterative, so a path with hundreds of thousands of vertices does not consume the call stack.

API

struct BiconnectedComponentsResult {
    std::vector<std::vector<int>> components;
    std::vector<std::vector<int>> edge_components;
    std::vector<int> component_of_edge;
    std::vector<std::vector<int>> vertex_components;
    std::vector<int> articulation;
    std::vector<int> ord;
    std::vector<int> low;

    int component_count() const;
    bool is_articulation(int vertex) const;
};

template <class T>
BiconnectedComponentsResult biconnected_components(const Graph<T>& graph);

Member or function	Description	Complexity
`components[c]`	Vertices belonging to block `c`; their order is unspecified.	–
`edge_components[c]`	Active edge IDs belonging to block `c`. Singleton blocks have no edges.	–
`component_of_edge[e]`	Block containing edge `e`, or `-1` when edge `e` is inactive.	$O(1)$
`vertex_components[v]`	All blocks containing vertex `v`.	–
`articulation`	Articulation vertices in increasing vertex order.	–
`ord[v]`, `low[v]`	DFS order and lowlink value of vertex `v`.	$O(1)$
`component_count()`	Number of blocks, including isolated singleton blocks.	$O(1)$
`is_articulation(v)`	Whether `v` belongs to at least two blocks.	$O(1)$
`biconnected_components(graph)`	Computes the complete decomposition.	$O(N + M)$

The result owns all returned vectors and does not mutate the graph. Across all components, vertex incidences and stored edge IDs use $O(N + M)$ memory.

Example

#include "graph/biconnected_components.hpp"
#include "graph/graph.hpp"

#include <iostream>

int main() {
    m1une::graph::Graph<> graph(5);
    graph.add_edge(0, 1);
    graph.add_edge(1, 2);
    graph.add_edge(2, 0);
    graph.add_edge(1, 3);
    graph.add_edge(3, 4);

    auto result = m1une::graph::biconnected_components(graph);
    std::cout << result.component_count() << "\n";  // 3
    std::cout << result.is_articulation(1) << "\n";  // 1
    std::cout << result.is_articulation(3) << "\n";  // 1
}

Depends on

Graph (graph/graph.hpp)

Required by

Verified with

Code

#ifndef M1UNE_GRAPH_BICONNECTED_COMPONENTS_HPP
#define M1UNE_GRAPH_BICONNECTED_COMPONENTS_HPP 1

#include <cassert>
#include <vector>

#include "graph.hpp"

namespace m1une {
namespace graph {

struct BiconnectedComponentsResult {
    std::vector<std::vector<int>> components;
    std::vector<std::vector<int>> edge_components;
    std::vector<int> component_of_edge;
    std::vector<std::vector<int>> vertex_components;
    std::vector<int> articulation;
    std::vector<int> ord;
    std::vector<int> low;

    int component_count() const {
        return int(components.size());
    }

    bool is_articulation(int vertex) const {
        assert(0 <= vertex && vertex < int(vertex_components.size()));
        return vertex_components[vertex].size() >= 2;
    }
};

// Decomposes an undirected graph into maximal vertex-biconnected blocks.
// Every active edge belongs to exactly one block. Isolated vertices form
// singleton blocks, and articulation vertices occur in multiple blocks.
template <class T>
BiconnectedComponentsResult biconnected_components(const Graph<T>& graph) {
    const int n = graph.size();
    const int edge_count = graph.edge_count();

    BiconnectedComponentsResult result;
    result.component_of_edge.assign(edge_count, -1);
    result.vertex_components.assign(n, {});
    result.ord.assign(n, -1);
    result.low.assign(n, -1);

    std::vector<int> edge_from(edge_count, -1);
    std::vector<int> edge_to(edge_count, -1);
    std::vector<int> incidence_count(edge_count, 0);
    std::vector<int> alive_degree(n, 0);
    for (int vertex = 0; vertex < n; vertex++) {
        for (const Edge<T>& edge : graph[vertex]) {
            if (!edge.alive) continue;
            assert(0 <= edge.id && edge.id < edge_count);
            alive_degree[vertex]++;
            if (incidence_count[edge.id] == 0) {
                edge_from[edge.id] = edge.from;
                edge_to[edge.id] = edge.to;
            }
            incidence_count[edge.id]++;
        }
    }
#ifndef NDEBUG
    for (int edge_id = 0; edge_id < edge_count; edge_id++) {
        if (incidence_count[edge_id] == 0) continue;
        assert(incidence_count[edge_id] == 2);
        assert(edge_from[edge_id] != edge_to[edge_id]);
    }
#endif

    std::vector<int> parent(n, -1);
    std::vector<int> parent_edge(n, -1);
    std::vector<int> next_edge(n, 0);
    std::vector<int> dfs_stack;
    std::vector<int> edge_stack;
    std::vector<int> vertex_mark(n, -1);
    int timer = 0;

    auto add_singleton = [&](int vertex) {
        const int component = result.component_count();
        result.components.push_back(std::vector<int>(1, vertex));
        result.edge_components.emplace_back();
        result.vertex_components[vertex].push_back(component);
    };

    auto extract_component = [&](int stopping_edge) {
        const int component = result.component_count();
        result.components.emplace_back();
        result.edge_components.emplace_back();
        std::vector<int>& vertices = result.components.back();
        std::vector<int>& edges = result.edge_components.back();

        while (true) {
            assert(!edge_stack.empty());
            const int edge_id = edge_stack.back();
            edge_stack.pop_back();
            edges.push_back(edge_id);
            result.component_of_edge[edge_id] = component;

            const int endpoints[2] = {edge_from[edge_id], edge_to[edge_id]};
            for (int vertex : endpoints) {
                if (vertex_mark[vertex] == component) continue;
                vertex_mark[vertex] = component;
                vertices.push_back(vertex);
            }
            if (edge_id == stopping_edge) break;
        }
        for (int vertex : vertices) {
            result.vertex_components[vertex].push_back(component);
        }
    };

    for (int root = 0; root < n; root++) {
        if (result.ord[root] != -1) continue;
        if (alive_degree[root] == 0) {
            result.ord[root] = result.low[root] = timer++;
            add_singleton(root);
            continue;
        }

        result.ord[root] = result.low[root] = timer++;
        dfs_stack.push_back(root);
        while (!dfs_stack.empty()) {
            const int vertex = dfs_stack.back();
            if (next_edge[vertex] < int(graph[vertex].size())) {
                const Edge<T>& edge = graph[vertex][next_edge[vertex]++];
                if (!edge.alive || edge.id == parent_edge[vertex]) continue;
                const int to = edge.to;
                if (result.ord[to] == -1) {
                    parent[to] = vertex;
                    parent_edge[to] = edge.id;
                    edge_stack.push_back(edge.id);
                    result.ord[to] = result.low[to] = timer++;
                    dfs_stack.push_back(to);
                } else if (result.ord[to] < result.ord[vertex]) {
                    edge_stack.push_back(edge.id);
                    if (result.ord[to] < result.low[vertex]) {
                        result.low[vertex] = result.ord[to];
                    }
                }
                continue;
            }

            dfs_stack.pop_back();
            const int parent_vertex = parent[vertex];
            if (parent_vertex == -1) {
                assert(edge_stack.empty());
                continue;
            }
            if (result.low[vertex] < result.low[parent_vertex]) {
                result.low[parent_vertex] = result.low[vertex];
            }
            if (result.ord[parent_vertex] <= result.low[vertex]) {
                extract_component(parent_edge[vertex]);
            }
        }
    }

    for (int vertex = 0; vertex < n; vertex++) {
        if (result.is_articulation(vertex)) result.articulation.push_back(vertex);
    }
    return result;
}

}  // namespace graph
}  // namespace m1une

#endif  // M1UNE_GRAPH_BICONNECTED_COMPONENTS_HPP

#line 1 "graph/biconnected_components.hpp"



#include <cassert>
#include <vector>

#line 1 "graph/graph.hpp"



#include <array>
#line 6 "graph/graph.hpp"
#include <utility>
#line 8 "graph/graph.hpp"

namespace m1une {
namespace graph {

template <class T = int>
struct Edge {
    using cost_type = T;

    int from;
    int to;
    T cost;
    int id;
    bool alive;

    Edge() : from(-1), to(-1), cost(T()), id(-1), alive(true) {}
    Edge(int from_, int to_, T cost_ = T(1), int id_ = -1, bool alive_ = true)
        : from(from_), to(to_), cost(cost_), id(id_), alive(alive_) {}

    int other(int v) const {
        assert(v == from || v == to);
        return from ^ to ^ v;
    }
};

template <class T = int>
struct Graph {
    using edge_type = Edge<T>;
    using cost_type = T;

   private:
    struct EdgePositions {
        std::array<std::pair<int, int>, 2> value{};
        int size = 0;

        void push_back(std::pair<int, int> position) {
            assert(size < 2);
            value[size++] = position;
        }
    };

    int _n;
    int _edge_count;
    std::vector<std::vector<edge_type>> _g;
    std::vector<EdgePositions> _edge_positions;

   public:
    Graph() : _n(0), _edge_count(0) {}
    explicit Graph(int n) : _n(n), _edge_count(0), _g(n) {
        assert(0 <= n);
    }

    int size() const {
        return _n;
    }

    bool empty() const {
        return _n == 0;
    }

    int edge_count() const {
        return _edge_count;
    }

    int add_vertex() {
        _g.emplace_back();
        return _n++;
    }

    int add_directed_edge(int from, int to, T cost = T(1)) {
        assert(0 <= from && from < _n);
        assert(0 <= to && to < _n);
        int id = _edge_count++;
        int idx = int(_g[from].size());
        _g[from].push_back(edge_type(from, to, cost, id));
        _edge_positions.emplace_back();
        _edge_positions.back().push_back({from, idx});
        return id;
    }

    int add_edge(int u, int v, T cost = T(1)) {
        assert(0 <= u && u < _n);
        assert(0 <= v && v < _n);
        int id = _edge_count++;
        int u_idx = int(_g[u].size());
        _g[u].push_back(edge_type(u, v, cost, id));
        int v_idx = int(_g[v].size());
        _g[v].push_back(edge_type(v, u, cost, id));
        _edge_positions.emplace_back();
        _edge_positions.back().push_back({u, u_idx});
        _edge_positions.back().push_back({v, v_idx});
        return id;
    }

    void set_edge_alive(int id, bool alive) {
        assert(0 <= id && id < _edge_count);
        for (int i = 0; i < _edge_positions[id].size; ++i) {
            auto [v, idx] = _edge_positions[id].value[i];
            _g[v][idx].alive = alive;
        }
    }

    void erase_edge(int id) {
        set_edge_alive(id, false);
    }

    void revive_edge(int id) {
        set_edge_alive(id, true);
    }

    bool is_edge_alive(int id) const {
        assert(0 <= id && id < _edge_count);
        assert(_edge_positions[id].size != 0);
        auto [v, idx] = _edge_positions[id].value[0];
        return _g[v][idx].alive;
    }

    const std::vector<edge_type>& operator[](int v) const {
        assert(0 <= v && v < _n);
        return _g[v];
    }

    std::vector<edge_type>& operator[](int v) {
        assert(0 <= v && v < _n);
        return _g[v];
    }

    const std::vector<std::vector<edge_type>>& adjacency() const {
        return _g;
    }

    std::vector<std::vector<edge_type>>& adjacency() {
        return _g;
    }

    std::vector<edge_type> edges(bool include_inactive = false) const {
        std::vector<edge_type> result;
        result.reserve(_edge_count);
        std::vector<char> used(_edge_count, false);
        for (int v = 0; v < _n; v++) {
            for (const auto& e : _g[v]) {
                if (!include_inactive && !e.alive) continue;
                if (0 <= e.id && e.id < _edge_count) {
                    if (used[e.id]) continue;
                    used[e.id] = true;
                }
                result.push_back(e);
            }
        }
        return result;
    }

    Graph reversed() const {
        Graph result(_n);
        result._edge_count = _edge_count;
        result._edge_positions.assign(_edge_count, {});
        for (int v = 0; v < _n; v++) {
            for (const auto& e : _g[v]) {
                int idx = int(result._g[e.to].size());
                result._g[e.to].push_back(edge_type(e.to, e.from, e.cost, e.id, e.alive));
                if (0 <= e.id && e.id < _edge_count) result._edge_positions[e.id].push_back({e.to, idx});
            }
        }
        return result;
    }
};

}  // namespace graph
}  // namespace m1une


#line 8 "graph/biconnected_components.hpp"

namespace m1une {
namespace graph {

struct BiconnectedComponentsResult {
    std::vector<std::vector<int>> components;
    std::vector<std::vector<int>> edge_components;
    std::vector<int> component_of_edge;
    std::vector<std::vector<int>> vertex_components;
    std::vector<int> articulation;
    std::vector<int> ord;
    std::vector<int> low;

    int component_count() const {
        return int(components.size());
    }

    bool is_articulation(int vertex) const {
        assert(0 <= vertex && vertex < int(vertex_components.size()));
        return vertex_components[vertex].size() >= 2;
    }
};

// Decomposes an undirected graph into maximal vertex-biconnected blocks.
// Every active edge belongs to exactly one block. Isolated vertices form
// singleton blocks, and articulation vertices occur in multiple blocks.
template <class T>
BiconnectedComponentsResult biconnected_components(const Graph<T>& graph) {
    const int n = graph.size();
    const int edge_count = graph.edge_count();

    BiconnectedComponentsResult result;
    result.component_of_edge.assign(edge_count, -1);
    result.vertex_components.assign(n, {});
    result.ord.assign(n, -1);
    result.low.assign(n, -1);

    std::vector<int> edge_from(edge_count, -1);
    std::vector<int> edge_to(edge_count, -1);
    std::vector<int> incidence_count(edge_count, 0);
    std::vector<int> alive_degree(n, 0);
    for (int vertex = 0; vertex < n; vertex++) {
        for (const Edge<T>& edge : graph[vertex]) {
            if (!edge.alive) continue;
            assert(0 <= edge.id && edge.id < edge_count);
            alive_degree[vertex]++;
            if (incidence_count[edge.id] == 0) {
                edge_from[edge.id] = edge.from;
                edge_to[edge.id] = edge.to;
            }
            incidence_count[edge.id]++;
        }
    }
#ifndef NDEBUG
    for (int edge_id = 0; edge_id < edge_count; edge_id++) {
        if (incidence_count[edge_id] == 0) continue;
        assert(incidence_count[edge_id] == 2);
        assert(edge_from[edge_id] != edge_to[edge_id]);
    }
#endif

    std::vector<int> parent(n, -1);
    std::vector<int> parent_edge(n, -1);
    std::vector<int> next_edge(n, 0);
    std::vector<int> dfs_stack;
    std::vector<int> edge_stack;
    std::vector<int> vertex_mark(n, -1);
    int timer = 0;

    auto add_singleton = [&](int vertex) {
        const int component = result.component_count();
        result.components.push_back(std::vector<int>(1, vertex));
        result.edge_components.emplace_back();
        result.vertex_components[vertex].push_back(component);
    };

    auto extract_component = [&](int stopping_edge) {
        const int component = result.component_count();
        result.components.emplace_back();
        result.edge_components.emplace_back();
        std::vector<int>& vertices = result.components.back();
        std::vector<int>& edges = result.edge_components.back();

        while (true) {
            assert(!edge_stack.empty());
            const int edge_id = edge_stack.back();
            edge_stack.pop_back();
            edges.push_back(edge_id);
            result.component_of_edge[edge_id] = component;

            const int endpoints[2] = {edge_from[edge_id], edge_to[edge_id]};
            for (int vertex : endpoints) {
                if (vertex_mark[vertex] == component) continue;
                vertex_mark[vertex] = component;
                vertices.push_back(vertex);
            }
            if (edge_id == stopping_edge) break;
        }
        for (int vertex : vertices) {
            result.vertex_components[vertex].push_back(component);
        }
    };

    for (int root = 0; root < n; root++) {
        if (result.ord[root] != -1) continue;
        if (alive_degree[root] == 0) {
            result.ord[root] = result.low[root] = timer++;
            add_singleton(root);
            continue;
        }

        result.ord[root] = result.low[root] = timer++;
        dfs_stack.push_back(root);
        while (!dfs_stack.empty()) {
            const int vertex = dfs_stack.back();
            if (next_edge[vertex] < int(graph[vertex].size())) {
                const Edge<T>& edge = graph[vertex][next_edge[vertex]++];
                if (!edge.alive || edge.id == parent_edge[vertex]) continue;
                const int to = edge.to;
                if (result.ord[to] == -1) {
                    parent[to] = vertex;
                    parent_edge[to] = edge.id;
                    edge_stack.push_back(edge.id);
                    result.ord[to] = result.low[to] = timer++;
                    dfs_stack.push_back(to);
                } else if (result.ord[to] < result.ord[vertex]) {
                    edge_stack.push_back(edge.id);
                    if (result.ord[to] < result.low[vertex]) {
                        result.low[vertex] = result.ord[to];
                    }
                }
                continue;
            }

            dfs_stack.pop_back();
            const int parent_vertex = parent[vertex];
            if (parent_vertex == -1) {
                assert(edge_stack.empty());
                continue;
            }
            if (result.low[vertex] < result.low[parent_vertex]) {
                result.low[parent_vertex] = result.low[vertex];
            }
            if (result.ord[parent_vertex] <= result.low[vertex]) {
                extract_component(parent_edge[vertex]);
            }
        }
    }

    for (int vertex = 0; vertex < n; vertex++) {
        if (result.is_articulation(vertex)) result.articulation.push_back(vertex);
    }
    return result;
}

}  // namespace graph
}  // namespace m1une

m1une's library

This documentation is automatically generated by online-judge-tools/verification-helper

Biconnected Components
(graph/biconnected_components.hpp)

Overview

Graph Requirements

API

Example

Depends on

Required by

Verified with

Code

Biconnected Components (graph/biconnected_components.hpp)

Overview

Graph Requirements

API

Example

Depends on

Required by

Verified with

Code

Biconnected Components
(graph/biconnected_components.hpp)