• Undirected graph
  • Cut the edge,Cut point,Double dot,Side double
• Digraph
  • Strong connectivity component,Example

Undirected graph

• concept

  • time stamp
    \(dfn[x] \), in depth first traversal, integer marks are made in the order in which each node is first accessed
    

  • Retroactive value
    \(low[x] \), the minimum timestamp that can be reached by a non searching edge
    

• Edge cutting criterion

  • An undirected edge \ ((x,y) \) is a cut edge / bridge if and only if there is a child node of X satisfying \ (DFN [x] < low [y] \)
    After deleting the undirected edge ((x,y)), the graph is broken into two parts
    

  • board

int dfn[N], low[N], dfcnt;
bool g[M];
void tarjan(int x, int ei) {
    dfn[x] = low[x] = ++dfcnt;
    for(int i = head[x]; i; i = e[i].next) {
        int y = e[i].t;
        if (!dfn[y]) {
            tarjan(y, i);
            low[x] = min(low[x], low[y]);
            if (dfn[x] < low[y]) g[i] = g[i^1] = 1;
        }
        else if (i != (ei^1)) low[x] = min(low[x], dfn[y]);
    }
}

• Cut point decision rule

  • If x is not a root node, then x is a cut point if and only if there is a child node y satisfying \ (DFN [x] \ \ Leq low [y] \)
    If x is the root node, then x is the cut point if and only if there are at least two child nodes (y)_ 1,y_ 2)
    

  • board

int dfn[N], low[N], dfcnt, rt;
bool g[N];
void tarjan(int x) {
    dfn[x] = low[x] = ++dfcnt;
    int son = 0;
    for (int i = head[x]; i; i = e[i].next) {
        int y = e[i].t;
        if (!dfn[y]) {
            tarjan(y);
            low[x] = min(low[x], low[y]);
            if (dfn[x] <= low[y]) {
                son++;
                if (x != rt || son > 1) g[x] = 1;
            }
        }
        else low[x] = min(low[x], dfn[y]);
    }
}

• Point double connected component

  • For every double center, there is no cut point in the graph

  • board

int dfn[N], low[N], dfcnt, sta[N], top, cnt;
vector<int> dcc[N];
bool  g[N];
void tarjan(int x, int rt) {
    dfn[x] = low[x] = ++dfcnt;
    sta[++top] = x;
    int son = 0;
    for (int i = head[x]; i; i = e[i].next) {
        int y = e[i].t;
        if (!dfn[y]) {
            tarjan(y); 
            low[x] = min(low[x], low[y]);
            if (dfn[x] <= low[y]) {
                son++;
                if (x != rt || son > 1)  g[x] = 1;
                dcc[++cnt].clear();
                while (1) {
                    int z = sta[top--];
                    dcc[cnt].push_back(z);
                    if (y == z) break;
                }
                dcc[cnt].push_back(x);
            }
        }
        else low[x] = min(low[x], dfn[y]);
    }
}

• Side double connected component

  • For an edge double, any two points have two non coincident paths

  • board

int dfn[N], low[N], dfcnt;
bool g[M];
void tarjan(int x, int ei) {
    dfn[x] = low[x] = ++dfcnt;
    for(int i = head[x]; i; i = e[i].next) {
        int y = e[i].t;
        if (!dfn[y]) {
            tarjan(y, i);
            low[x] = min(low[x], low[y]);
            if (dfn[x] < low[y]) g[i] = g[i^1] = 1;
        }
        else if (i != (ei^1)) low[x] = min(low[x], dfn[y]);
    }
}
int n, m, d[N], b[N], cnt, ans;
void dfs(int x) {
    b[x] = cnt;
    for(int i = head[x]; i; i = e[i].next) {
        int y = e[i].t;
        if (b[y] || g[i]) continue;
        dfs(y);
    }
}
int main() {
    //~~~
    for(int i = 1; i <= n; i++)
        if (!dfn[i]) tarjan(i, 0);
    for(int i = 1; i <= n; i++)
        if (!b[i]) cnt++, dfs(i);
    //~~~
    return 0;
}

Digraph

• Strong connected components of Digraphs

  • In a strong connectivity component, if there is a path from x to y, there is a path from y to x

  • board

void tarjan(int x) {
    dfn[x] = low[x] = ++dfcnt;
    s[++top] = x;
    for(int i = head[x]; i; i = e[i].next) {
        int y = e[i].t;
        if (!dfn[y]) tarjan(y), low[x] = min(low[x], low[y]);
        else if (!b[y]) low[x] = min(low[x], dfn[y]);
    }
    if (dfn[x] == low[x]) {
        cnt++;
        while(1) {
            int y = s[top--];
            b[y] = cnt;
            size[cnt]++;
            if (x == y) break;
        }
    }
}

Example

• luoguP3387Contraction point

  • code

#include <queue>
#include <cstdio>
#include <cstring>
#include <algorithm>
using namespace std;
const int N = 1e4+5, M = 1e5+5;
struct side { int t, next; } e[M][2];
int head[N][2], tot[2];
void add(int x, int y, int k) {
    e[++tot[k]][k].next = head[x][k]; 
    head[x][k] = tot[k];
    e[tot[k]][k].t = y;
}
int n, m, w[N], r[N], d[N], ans;
int dfn[N], low[N], dfcnt, sta[N], top, cnt, bel[N], sum[N];
void tarjan(int x) {
    dfn[x] = low[x] = ++dfcnt;
    sta[++top] = x;
    for (int i = head[x][0]; i; i = e[i][0].next) {
        int y = e[i][0].t;
        if (!dfn[y]) tarjan(y), low[x] = min(low[x], low[y]);
        else if (!bel[y]) low[x] = min(low[x], dfn[y]);
    }
    if (dfn[x] == low[x]) {
        cnt++;
        while (1) {
            int y = sta[top--];
            bel[y] = cnt;
            sum[cnt] += w[y];
            if (x == y) break;
        }
    }
}
queue<int> q;
int tuopu() {
    for (int i = 1; i <= cnt; i++)
        if (!r[i]) q.push(i), d[i] = sum[i];
    while (!q.empty()) {
        int x = q.front(); q.pop();
        for (int i = head[x][1]; i; i = e[i][1].next) {
            int y = e[i][1].t;
            d[y] = max(d[y], d[x] + sum[y]);
            if (--r[y] == 0) q.push(y);
        }
    }
    for (int i = 1; i <= cnt; i++)
        ans = max(ans, d[i]);
    return ans;
}
int main() {
    scanf("%d%d", &n, &m);
    for (int i = 1; i <= n; i++)
        scanf("%d", &w[i]);
    for (int i = 1; i <= m; i++) {
        int x, y;
        scanf("%d%d", &x, &y);
        add(x, y, 0);
    }
    for (int i = 1; i <= n; i++)
        if (!dfn[i]) tarjan(i);
    for (int x = 1; x <= n; x++)
        for (int i = head[x][0]; i; i = e[i][0].next) {
            int y = e[i][0].t;
            if (bel[x] != bel[y]) 
                r[bel[y]]++, add(bel[x], bel[y], 1);
        }
    printf("%d\n", tuopu());
    return 0;
}