tarjan

桥

low[v]>dfn[u]  //成立条件

炸铁路

#include<bits/stdc++.h>
using namespace std;
#define ll long long 
const ll maxn=1e6+10;
ll n,m,cnt=0,ct=0,ti=0;
ll fa[maxn],head[maxn],dfn[maxn],low[maxn];
struct edge{
	ll u,v,nex;
}e[maxn],ans[maxn];
ll cmp(edge x,edge y){
	if(x.u!=y.u) return x.u<y.u;
	return x.v<y.v;
}

void add(ll u,ll v){
	cnt++;
	e[cnt].v=v;
	e[cnt].nex=head[u];
	head[u]=cnt;
}
void addedge(ll u,ll v){
	if(u>v){
		ll t=u;
		u=v;
		v=t;
	}
	ct++;
	ans[ct].u=u;
	ans[ct].v=v;
}
void tarjan(ll u){
	dfn[u]=low[u]=++ti;
	for(ll i=head[u];i;i=e[i].nex){
		ll v=e[i].v;
		if(dfn[v]&&fa[u]!=v) low[u]=min(low[u],dfn[v]);
		if(!dfn[v]){
			fa[v]=u;
			tarjan(v);
			low[u]=min(low[u],low[v]);
			if(low[v]>dfn[u]) addedge(u,v);	
		}
	}
}
int main()
{
	cin>>n>>m;
	for(ll i=1;i<=m;i++){
		ll u,v;
		cin>>u>>v;
		add(u,v);
		add(v,u);
	}
	tarjan(1);
	sort(ans+1,ans+1+ct,cmp);
	for(ll i=1;i<=ct;i++){
		cout<<ans[i].u<<' '<<ans[i].v<<endl;
	}
	return 0;
}

割点

low[v]>=dfn[u] //子节点条件
isroot==u&&tot>=2 //是根节点且子节点个数大于等于2
isroot!=u&&tot>=1 //不是根节点且子节点个数大于等于1

P3388 【模板】割点（割顶）

#include<bits/stdc++.h>
using namespace std;
#define ll long long 
const ll maxn=1e5+10;
ll head[maxn],dfn[maxn],low[maxn],ans[maxn];
ll n,m,isroot,cnt=0,idx=0;
ll ti=0;
struct edge{
	ll v,nex;
}e[maxn<<1];
void add(ll u,ll v){
	cnt++;
	e[cnt].v=v;
	e[cnt].nex=head[u];
	head[u]=cnt;
}
void tarjan(ll u){
	ll tot=0;
	dfn[u]=low[u]=++ti;
	for(ll i=head[u];i;i=e[i].nex){
		ll v=e[i].v;
		if(!dfn[v]){
			tarjan(v);
			low[u]=min(low[u],low[v]);
			if(low[v]>=dfn[u]) tot++;
		}else {
			low[u]=min(low[u],dfn[v]);
		}
	}
	if( (isroot==u&&tot>=2)||(isroot!=u&&tot>=1) ){
		idx++;
		ans[idx]=u;
	}
}
int main()
{
	cin>>n>>m;
	for(ll i=1;i<=m;i++){
		ll u,v;
		cin>>u>>v;
		add(u,v);
		add(v,u);
	}
//	tarjan(1);
	for(ll i=1;i<=n;i++){
		if(dfn[i]==0) {
			isroot=i;
			tarjan(i);
		}
	}
	sort(ans+1,ans+1+idx);
	cout<<idx<<endl;
	for(ll i=1;i<=idx;i++){
		cout<<ans[i];
		if(i!=idx) cout<<' ';
		else cout<<endl;
	}
	return 0;
}

缩点

P3387 【模板】缩点

#include<bits/stdc++.h>
using namespace std;
#define ll long long
const ll maxn=1e5+10;
ll n,m;
ll dfn[maxn],low[maxn];
ll w[maxn],ww[maxn];
ll head[maxn],hd[maxn];
ll rep[maxn];
ll cnt=0,cntt=0;
ll tim=0;
ll rudu[maxn];
ll ans=0;
ll dist[maxn];
ll vis[maxn];
struct edge{
	ll v,nex,u;
}e[maxn],ed[maxn];
void add(ll u,ll v){
	cnt++;
	e[cnt].u=u;
	e[cnt].v=v;
	e[cnt].nex=head[u];
	head[u]=cnt;
	return ;
}
void init(){
	memset(head,0,sizeof(head));
	memset(hd,0,sizeof(hd));
	memset(vis,0,sizeof(vis));
	return ;
}
stack<ll>st;
void tarjan(ll u){
	low[u]=dfn[u]=++tim;//时间戳 首次访问时间
	vis[u]=1;
	st.push(u);
	for(ll i=head[u];i;i=e[i].nex){
		ll v=e[i].v;
		if(dfn[v]==0){
			tarjan(v);
			low[u]=min(low[u],low[v]);
		}else if(vis[v]==1){
			low[u]=min(low[u],dfn[v]);
		}
	} 
//	cout<<"YES"<<endl;
	if(dfn[u]==low[u]){//出栈 
		while(st.empty()!=1){
			ll t =st.top();
			st.pop();
			rep[t]=u;
			if(u!=t)
			w[u]+=w[t];
			vis[t]=0;
			if(u==t){
				break;
			}
		}
	}
	return ;
}
queue<ll>q;
void toposort(){
	for(ll i=1;i<=n;i++){
		if(rep[i]==i&&rudu[i]==0){
			q.push(i);
			dist[i]=w[i];
		}
	}
	while(q.empty()!=1){
		ll u=q.front();
		q.pop();
		for(ll i=hd[u];i;i=ed[i].nex){
			ll v=ed[i].v;
			dist[v]=max(dist[v],dist[u]+w[v]);
			rudu[v]--;
			if(rudu[v]==0) q.push(v);
		}
	}
	for(ll i=1;i<=n;i++){
		if(rep[i]==i){
			ans=max(ans,dist[i]);
		}
	}
	return ;
}
void show(){
	for(ll i=1;i<=n;i++){
		cout<<w[i]<<' ';
	}
	cout<<endl;
	for(ll i=1;i<=n;i++){
		cout<<rep[i]<<' ';
	}
	cout<<endl;
}
int main()
{
	init();
	cin>>n>>m;
	for(ll i=1;i<=n;i++){
		cin>>w[i];
	}
	ll u,v;
	for(ll i=1;i<=m;i++){
		cin>>u>>v;
		add(u,v);
	}
	for(ll i=1;i<=n;i++){
		if(dfn[i]==0) tarjan(i);
	}
	for(ll i=1;i<=m;i++){
		ll x=rep[e[i].u];
		ll y=rep[e[i].v];
		if(x!=y){
			cntt++;
			ed[cntt].v=y;
			ed[cntt].u=x;
			ed[cntt].nex=hd[x];
			hd[x]=cntt;
			rudu[y]++;
		}
	}
//	show();
	toposort();
	
	cout<<ans<<endl;
	return 0;
 }