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tarjan-割点-割边（桥）1

HHhy140516
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@mioou3gf
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2025/12/02 22:43
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基础

割点（割顶）：删除一个点，图的联通分量增加，这个点就是割点。
割边（桥）：删除一个边，图的联通分量增加，这个边就是割点。

例题

P3388 【模板】割点（割顶）

对于儿子能会到的点大于等于现在的点，且儿子个数大于二也不是根节点就是。
CPP
//Code by hhy
#include<bits/stdc++.h>
#define F(i , a , b , c) for( int i = (a) ; ((c > 0) ? i <= (b) : i >= (b)) ; i += c )
#define T(i , root , b , c) for( int i = root ; b ; c )
#define int long long
#define W(f) while(f)
#define ull unsigned long long
#define pb push_back
#define fi first
#define se second
#define ll long long
#define debug(...){\
	cout<<"debug in function "<<__FUNCTION__<<",line "<<__LINE__<<"\n";\
	string s=#__VA_ARGS__,s2="";\
	vector<string>v;\
	for(auto i:s){\
		if(i==','){\
			v.push_back(s2);\
			s2="";\
		}else{\
			s2+=i;\
		}\
	}\
	v.push_back(s2);\
	vector<int>v2={__VA_ARGS__};\
	for(int i=0;i<v.size()-1;i++){\
		cout<<v[i]<<"="<<v2[i]<<"\n";\
	}\
	cout<<v[v.size()-1]<<"="<<v2[v2.size()-1]<<"\n\n";\
}
using namespace std ;
const int kMaxN = 2e5 + 5 , MOD = 998244353 , INF = 1e18 ;
struct Edgepr
{
	int u , w ;
};
struct Edgeve
{
	int v , w ;
};
struct node
{
  int v , nxt ;
}edge[kMaxN] ;
int n , m , tim , cnt , res , root ;
int color[kMaxN] , dfn[kMaxN] , low[kMaxN] , head[kMaxN] ;
bool vis[kMaxN] ;
inline ll ksm(ll a , ll b)
{
	ll mul = 1 ;
	while(b)
	{
		if(b & 1) mul *= a , mul %= MOD ;
		a *= a ;
		a %= MOD ;
		b >>= 1 ;
	}
	return mul ;
}
inline int read()
{
    int x = 0 , f = 1 ;
    char ch = getchar() ;
    while(ch < '0' || ch > '9')
    {
        if(ch == '-') f = -1 ;
        ch = getchar() ;
    }
    while(ch >= '0' && ch <= '9') x = x * 10 + ch - '0' , ch = getchar() ;
    return x * f ;
}
void write(int x)
{
    if(x < 0) putchar('-') , x = -x ;
    if(x > 9) write(x / 10) ;
    putchar(x % 10 + '0') ;
}
void tanjan(int u)
{
	dfn[u] = low[u] = ++tim ;
	int ans = 0 ;
	for( int i = head[u] ; i ; i = edge[i].nxt )
	{
		int v = edge[i].v ;
		if(dfn[v] == 0) 
		{
			tanjan(v) ;
			low[u] = min(low[u] , low[v]) ;
			if(low[v] >= dfn[u])
			{
				ans++ ;
				if(u != root || ans >= 2) vis[u] = 1 ;
			}
		}
		else
		{
			low[u] = min(low[u] , dfn[v]) ;
		}
	}
}
void add_edge(int u , int v)
{
	edge[++cnt] = {v , head[u]} ;
	head[u] = cnt ;
}
void work()
{
	cin >> n >> m ;
	for( int i = 1 ; i <= m ; i++ )
	{
		int u , v ;
		cin >> u >> v ;
		add_edge(u , v) ;
		add_edge(v , u) ;
	}
	for( int i = 1 ; i <= n ; i++ )
	{
		if(dfn[i] == 0) 
		{
			root = i ;
			tanjan(i) ;
		}
	}
	for( int i = 1 ; i <= n ; i++ )
	{
		if(vis[i]) res++ ;
	}
	cout << res << "\n" ;
	for( int i = 1 ; i <= n ; i++ )
	{
		if(vis[i])
		{
			cout << i << " " ;
		}
	}
}
signed main()
{
//	freopen(".in" , "r" , stdin) ;
//	freopen(".out" , "w" , stdout) ;
	ios::sync_with_stdio(0) , cin.tie(0) , cout.tie(0) ;
  int t = 1 ;
  // cin >> t ;
  while(t--) work() ;
  return 0 ;
}

HDU4738 Caocao's Bridges

就是求割点的边的最小值即可。
CPP
//Code by hhy
#include<bits/stdc++.h>
#define F(i , a , b , c) for( int i = (a) ; ((c > 0) ? i <= (b) : i >= (b)) ; i += c )
#define T(i , root , b , c) for( int i = root ; b ; c )
#define int long long
#define W(f) while(f)
#define ull unsigned long long
#define pb push_back
#define fi first
#define se second
#define ll long long
#define debug(...){\
	cout<<"debug in function "<<__FUNCTION__<<",line "<<__LINE__<<"\n";\
	string s=#__VA_ARGS__,s2="";\
	vector<string>v;\
	for(auto i:s){\
		if(i==','){\
			v.push_back(s2);\
			s2="";\
		}else{\
			s2+=i;\
		}\
	}\
	v.push_back(s2);\
	vector<int>v2={__VA_ARGS__};\
	for(int i=0;i<v.size()-1;i++){\
		cout<<v[i]<<"="<<v2[i]<<"\n";\
	}\
	cout<<v[v.size()-1]<<"="<<v2[v2.size()-1]<<"\n\n";\
}
using namespace std ;
const int kMaxN = 1e3 + 5 , MOD = 998244353 , INF = 1e18 ;
struct Edgepr
{
	int u , w ;
};
struct Edgeve
{
	int v , w ;
};
struct node
{
  int v , w , nxt ;
}edge[kMaxN * kMaxN * 2] ;
int n , m , tim , cnt , ans ;
int dfn[kMaxN] , low[kMaxN] , head[kMaxN] ;
inline ll ksm(ll a , ll b)
{
	ll mul = 1 ;
	while(b)
	{
		if(b & 1) mul *= a , mul %= MOD ;
		a *= a ;
		a %= MOD ;
		b >>= 1 ;
	}
	return mul ;
}
inline int read()
{
    int x = 0 , f = 1 ;
    char ch = getchar() ;
    while(ch < '0' || ch > '9')
    {
        if(ch == '-') f = -1 ;
        ch = getchar() ;
    }
    while(ch >= '0' && ch <= '9') x = x * 10 + ch - '0' , ch = getchar() ;
    return x * f ;
}
void write(int x)
{
    if(x < 0) putchar('-') , x = -x ;
    if(x > 9) write(x / 10) ;
    putchar(x % 10 + '0') ;
}
void tanjan(int u , int fa)
{
	dfn[u] = low[u] = ++tim ;
	for( int i = head[u] ; i ; i = edge[i].nxt )
	{
		int v = edge[i].v ;
		if(dfn[v] == 0) 
		{
			tanjan(v , i ^ 1) ;
			low[u] = min(low[u] , low[v]) ;
			if(low[v] > dfn[u])
			{
				ans = min(ans , edge[i].w);
			}
		}
		else if(i != fa)
		{
			low[u] = min(low[u] , dfn[v]) ;
		}
	}
}
void add_edge(int u , int v , int w)
{
	edge[++cnt] = {v , w , head[u]} ;
	head[u] = cnt ;
}
void work()
{
	while(cin >> n >> m && n)
	{
		tim = cnt = ans = 0 ;
		cnt++ ;
		memset(head , 0 , sizeof head) ;
		memset(dfn , 0 , sizeof dfn) ;
		memset(low , 0 , sizeof low) ;
		for( int i = 1 ; i <= m ; i++ )
		{
			int u , v , w ;
			cin >> u >> v >> w ;
			add_edge(u , v , w) ;
			add_edge(v , u , w) ;
		}
		ans = INF ;
		tanjan(1 , 0) ;
		if(ans == INF) cout << "-1\n" ;
		else if(tim != n) cout << "0\n" ;
		else if(ans == 0) cout << "1\n" ;
		else cout << ans << "\n" ;
	}
}
signed main()
{
//	freopen(".in" , "r" , stdin) ;
//	freopen(".out" , "w" , stdout) ;
	ios::sync_with_stdio(0) , cin.tie(0) , cout.tie(0) ;
  int t = 1 ;
  // cin >> t ;
  while(t--) work() ;
  return 0 ;
}

U132350 【模板】割边（桥）

和割点相似。
CPP
//Code by hhy
#include<bits/stdc++.h>
#define F(i , a , b , c) for( int i = (a) ; ((c > 0) ? i <= (b) : i >= (b)) ; i += c )
#define T(i , root , b , c) for( int i = root ; b ; c )
#define int long long
#define W(f) while(f)
#define ull unsigned long long
#define pb push_back
#define fi first
#define se second
#define ll long long
#define debug(...){\
	cout<<"debug in function "<<__FUNCTION__<<",line "<<__LINE__<<"\n";\
	string s=#__VA_ARGS__,s2="";\
	vector<string>v;\
	for(auto i:s){\
		if(i==','){\
			v.push_back(s2);\
			s2="";\
		}else{\
			s2+=i;\
		}\
	}\
	v.push_back(s2);\
	vector<int>v2={__VA_ARGS__};\
	for(int i=0;i<v.size()-1;i++){\
		cout<<v[i]<<"="<<v2[i]<<"\n";\
	}\
	cout<<v[v.size()-1]<<"="<<v2[v2.size()-1]<<"\n\n";\
}
using namespace std ;
const int kMaxN = 2e5 + 5 , MOD = 998244353 , INF = 1e18 ;
struct Edgepr
{
	int u , w ;
};
struct Edgeve
{
	int v , w ;
};
struct node
{
  int v , nxt ;
}edge[kMaxN] ;
int n , m , tim , cnt , res ;
int color[kMaxN] , dfn[kMaxN] , low[kMaxN] , head[kMaxN] ;
bool vis[kMaxN] ;
inline ll ksm(ll a , ll b)
{
	ll mul = 1 ;
	while(b)
	{
		if(b & 1) mul *= a , mul %= MOD ;
		a *= a ;
		a %= MOD ;
		b >>= 1 ;
	}
	return mul ;
}
inline int read()
{
    int x = 0 , f = 1 ;
    char ch = getchar() ;
    while(ch < '0' || ch > '9')
    {
        if(ch == '-') f = -1 ;
        ch = getchar() ;
    }
    while(ch >= '0' && ch <= '9') x = x * 10 + ch - '0' , ch = getchar() ;
    return x * f ;
}
void write(int x)
{
    if(x < 0) putchar('-') , x = -x ;
    if(x > 9) write(x / 10) ;
    putchar(x % 10 + '0') ;
}
void tanjan(int u , int fa)
{
	dfn[u] = low[u] = ++tim ;
	for( int i = head[u] ; i ; i = edge[i].nxt )
	{
		int v = edge[i].v ;
		if(v == fa) continue ;
		if(dfn[v] == 0) 
		{
			tanjan(v , u) ;
			low[u] = min(low[u] , low[v]) ;
			if(low[v] > dfn[u])
			{
				res++ ;
			}
		}
		else
		{
			low[u] = min(low[u] , dfn[v]) ;
		}
	}
}
void add_edge(int u , int v)
{
	edge[++cnt] = {v , head[u]} ;
	head[u] = cnt ;
}
void work()
{
	cin >> n >> m ;
	for( int i = 1 ; i <= m ; i++ )
	{
		int u , v ;
		cin >> u >> v ;
		add_edge(u , v) ;
		add_edge(v , u) ;
	}
	for( int i = 1 ; i <= n ; i++ )
	{
		if(dfn[i] == 0) 
		{
			tanjan(i , 0) ;
		}
	}
	cout << res << "\n" ;
}
signed main()
{
//	freopen(".in" , "r" , stdin) ;
//	freopen(".out" , "w" , stdout) ;
	ios::sync_with_stdio(0) , cin.tie(0) , cout.tie(0) ;
  int t = 1 ;
  // cin >> t ;
  while(t--) work() ;
  return 0 ;
}

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