Block-Cut Tree
(graph/block_cut_tree.hpp)

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

Overview

This header converts a vertex-biconnected-components decomposition into its block-cut forest. Each block becomes one node, each articulation vertex becomes one node, and an edge joins a block to every articulation vertex it contains. For a connected input graph the result is a tree; for a disconnected graph it is a forest.

Every original vertex has a canonical node. An articulation vertex maps to its articulation node, while any other vertex maps to its unique block node. This is useful for reducing vertex-separator path queries to ordinary tree queries.

Graph Requirements

The same requirements as biconnected_components apply: build an undirected graph with Graph<T>::add_edge; self-loops are unsupported, parallel edges are supported, and inactive edges are ignored. Isolated vertices become isolated block nodes.

Node Numbering

Block nodes have IDs in [0, block_count()), in the same order as BiconnectedComponentsResult::components. Articulation nodes follow block nodes, in increasing order of the original vertex ID.

API

struct BlockCutTreeResult {
    std::vector<std::vector<int>> forest;
    std::vector<int> node_of_block;
    std::vector<int> node_of_articulation;
    std::vector<int> node_of_vertex;
    std::vector<int> block_of_node;
    std::vector<int> articulation_of_node;

    int node_count() const;
    int block_count() const;
    bool is_block_node(int node) const;
    bool is_articulation_node(int node) const;
};

BlockCutTreeResult block_cut_tree(
    const BiconnectedComponentsResult& biconnected
);

template <class T>
BlockCutTreeResult block_cut_tree(const Graph<T>& graph);

Member or function	Description	Complexity
`forest[node]`	Adjacent nodes in the block-cut forest.	–
`node_of_block[block]`	Forest node representing `block`.	$O(1)$
`node_of_articulation[v]`	Articulation node for original vertex `v`, or `-1` if `v` is not an articulation.	$O(1)$
`node_of_vertex[v]`	Canonical forest node representing original vertex `v`.	$O(1)$
`block_of_node[node]`	Original block ID, or `-1` for an articulation node.	$O(1)$
`articulation_of_node[node]`	Original articulation vertex, or `-1` for a block node.	$O(1)$
`node_count()`	Number of block and articulation nodes.	$O(1)$
`block_count()`	Number of block nodes.	$O(1)$
`is_block_node(node)`	Whether `node` represents a block.	$O(1)$
`is_articulation_node(node)`	Whether `node` represents an articulation vertex.	$O(1)$
`block_cut_tree(biconnected)`	Builds the forest from an existing decomposition.	$O(N + M)$
`block_cut_tree(graph)`	Computes the decomposition and builds the forest.	$O(N + M)$

The graph overload does not mutate the input. The result owns its vectors and uses $O(N + M)$ memory. Use the decomposition overload when the blocks are also needed, so they are not computed twice.

Example

#include "graph/block_cut_tree.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 biconnected = m1une::graph::biconnected_components(graph);
    auto block_cut = m1une::graph::block_cut_tree(biconnected);
    std::cout << block_cut.node_count() << "\n";  // 5
    std::cout << block_cut.node_of_articulation[1] << "\n";
    std::cout << block_cut.node_of_vertex[0] << "\n";
}

Depends on

Required by

Verified with

Code

#ifndef M1UNE_GRAPH_BLOCK_CUT_TREE_HPP
#define M1UNE_GRAPH_BLOCK_CUT_TREE_HPP 1

#include <cassert>
#include <vector>

#include "biconnected_components.hpp"

namespace m1une {
namespace graph {

struct BlockCutTreeResult {
    std::vector<std::vector<int>> forest;
    std::vector<int> node_of_block;
    std::vector<int> node_of_articulation;
    std::vector<int> node_of_vertex;
    std::vector<int> block_of_node;
    std::vector<int> articulation_of_node;

    int node_count() const {
        return int(forest.size());
    }

    int block_count() const {
        return int(node_of_block.size());
    }

    bool is_block_node(int node) const {
        assert(0 <= node && node < node_count());
        return block_of_node[node] != -1;
    }

    bool is_articulation_node(int node) const {
        assert(0 <= node && node < node_count());
        return articulation_of_node[node] != -1;
    }
};

// Builds the block-cut forest of a biconnected-components decomposition.
// Block nodes have IDs [0, block_count); articulation nodes follow them.
inline BlockCutTreeResult block_cut_tree(
    const BiconnectedComponentsResult& biconnected
) {
    const int vertex_count = int(biconnected.vertex_components.size());
    const int block_count = biconnected.component_count();

    BlockCutTreeResult result;
    result.node_of_block.resize(block_count);
    result.node_of_articulation.assign(vertex_count, -1);
    result.node_of_vertex.assign(vertex_count, -1);
    result.forest.resize(block_count);
    result.block_of_node.resize(block_count);
    result.articulation_of_node.assign(block_count, -1);
    for (int block = 0; block < block_count; block++) {
        result.node_of_block[block] = block;
        result.block_of_node[block] = block;
    }

    for (int vertex = 0; vertex < vertex_count; vertex++) {
        const std::vector<int>& blocks = biconnected.vertex_components[vertex];
        assert(!blocks.empty());
        if (blocks.size() == 1) {
            assert(0 <= blocks[0] && blocks[0] < block_count);
            result.node_of_vertex[vertex] = result.node_of_block[blocks[0]];
            continue;
        }

        const int node = result.node_count();
        result.node_of_articulation[vertex] = node;
        result.node_of_vertex[vertex] = node;
        result.forest.emplace_back();
        result.block_of_node.push_back(-1);
        result.articulation_of_node.push_back(vertex);
        for (int block : blocks) {
            assert(0 <= block && block < block_count);
            const int block_node = result.node_of_block[block];
            result.forest[node].push_back(block_node);
            result.forest[block_node].push_back(node);
        }
    }
    return result;
}

template <class T>
BlockCutTreeResult block_cut_tree(const Graph<T>& graph) {
    return block_cut_tree(biconnected_components(graph));
}

}  // namespace graph
}  // namespace m1une

#endif  // M1UNE_GRAPH_BLOCK_CUT_TREE_HPP

#line 1 "graph/block_cut_tree.hpp"



#include <cassert>
#include <vector>

#line 1 "graph/biconnected_components.hpp"



#line 6 "graph/biconnected_components.hpp"

#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


#line 8 "graph/block_cut_tree.hpp"

namespace m1une {
namespace graph {

struct BlockCutTreeResult {
    std::vector<std::vector<int>> forest;
    std::vector<int> node_of_block;
    std::vector<int> node_of_articulation;
    std::vector<int> node_of_vertex;
    std::vector<int> block_of_node;
    std::vector<int> articulation_of_node;

    int node_count() const {
        return int(forest.size());
    }

    int block_count() const {
        return int(node_of_block.size());
    }

    bool is_block_node(int node) const {
        assert(0 <= node && node < node_count());
        return block_of_node[node] != -1;
    }

    bool is_articulation_node(int node) const {
        assert(0 <= node && node < node_count());
        return articulation_of_node[node] != -1;
    }
};

// Builds the block-cut forest of a biconnected-components decomposition.
// Block nodes have IDs [0, block_count); articulation nodes follow them.
inline BlockCutTreeResult block_cut_tree(
    const BiconnectedComponentsResult& biconnected
) {
    const int vertex_count = int(biconnected.vertex_components.size());
    const int block_count = biconnected.component_count();

    BlockCutTreeResult result;
    result.node_of_block.resize(block_count);
    result.node_of_articulation.assign(vertex_count, -1);
    result.node_of_vertex.assign(vertex_count, -1);
    result.forest.resize(block_count);
    result.block_of_node.resize(block_count);
    result.articulation_of_node.assign(block_count, -1);
    for (int block = 0; block < block_count; block++) {
        result.node_of_block[block] = block;
        result.block_of_node[block] = block;
    }

    for (int vertex = 0; vertex < vertex_count; vertex++) {
        const std::vector<int>& blocks = biconnected.vertex_components[vertex];
        assert(!blocks.empty());
        if (blocks.size() == 1) {
            assert(0 <= blocks[0] && blocks[0] < block_count);
            result.node_of_vertex[vertex] = result.node_of_block[blocks[0]];
            continue;
        }

        const int node = result.node_count();
        result.node_of_articulation[vertex] = node;
        result.node_of_vertex[vertex] = node;
        result.forest.emplace_back();
        result.block_of_node.push_back(-1);
        result.articulation_of_node.push_back(vertex);
        for (int block : blocks) {
            assert(0 <= block && block < block_count);
            const int block_node = result.node_of_block[block];
            result.forest[node].push_back(block_node);
            result.forest[block_node].push_back(node);
        }
    }
    return result;
}

template <class T>
BlockCutTreeResult block_cut_tree(const Graph<T>& graph) {
    return block_cut_tree(biconnected_components(graph));
}

}  // namespace graph
}  // namespace m1une

m1une's library

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

Block-Cut Tree
(graph/block_cut_tree.hpp)

Overview

Graph Requirements

Node Numbering

API

Example

Depends on

Required by

Verified with

Code

Block-Cut Tree (graph/block_cut_tree.hpp)

Overview

Graph Requirements

Node Numbering

API

Example

Depends on

Required by

Verified with

Code

Block-Cut Tree
(graph/block_cut_tree.hpp)