Articulation points

2016-02-18 2024-05-07 #algorithms #data-structures #post


Hey guys! Currently I am training for a programming contest and I have no time to publish. I am learning many interesting topics about algorithms and programming and I want to share that with you. However, to write an article consumes much of my time and I’m devising a format that is both brief and useful: Algorithm or Data structure, brief info, sample problem, solution and references.

Articulation points

Today’s topic is about graph theory, articulation points. Here a brief info from Dave Mount:

Articulation Points and Biconnected Graphs: Today we discuss another application of DFS, this time to a problem on undirected graphs. Let G = (V, E) be a ** connected** undirected graph. Consider de following definitions.

Articulation Point( or Cut Vertex): Is any vertex whose removal(together with the removal of any incident edges) results in a disconnected graph.

Articulation points example

Sample problem

A problem where you can put in practice this topic is 1063 Ant Hills from

Solution to sample problem

#include <iostream>
#include <cstdio>
#include <vector>
#include <cstring>
#include <algorithm>

using namespace std;

const int LIMIT = 10010;

vector<int> G[LIMIT];
int D[LIMIT];
int P[LIMIT];
int L[LIMIT];
bool V[LIMIT];

int n, m, ap, order;

int art_points(int u)
    int i, v, found = 0, sum = 0;
    V[u] = true;
    L[u] = D[u] = order++;

    for (i = 0; i < int(G[u].size()); i++) {
        v = G[u][i];

        if (!V[v]) {
            P[v] = u;
            sum += art_points(v);
            L[u] = min(L[u], L[v]);

            if (P[u] == -1) { // Special case: root
                if (i > 0) {    // Two or more childs
                    found = 1;
            } else {
                if (L[v] >= D[u]) {
                    found = 1;
        } else if (v != P[u]) {
            L[u] = min(L[u], D[v]);

    return sum + found;

int main(int argc, char **argv)
    int t, tc, i, u, v;

    scanf("%d", &t);

    for (tc = 1; tc <= t; tc++) {
        scanf("%d %d", &n, &m);

        for (i = 0; i < m; i++) {
            scanf("%d %d", &u, &v);

        order = 0;
        ap = 0;
        P[1] = -1;

        printf("Case %d: %d\n", tc, art_points(1));

        if (tc < t) {
            memset(V, 0, sizeof V);
            memset(L, 0, sizeof L);
            memset(D, 0, sizeof D);
            memset(P, 0, sizeof P);

        for (i = 0; i <= n; i++)

    return 0;


1 Articulation Points and Biconnected Components
2 Articulation Points Detection Algorithm
3 Biconnected component