圆方树

圆方树是Tarjan发明，ImmortalCO引入CNOI界的一种处理无向图/仙人掌的利器

具体地，圆方树可以简便地把一个无向图变成树/仙人掌

做法就是Tarjan搜点双的时候，对于每个点双我们建一个方点向这个点双里的点连边，割点向相邻的方点连边

然后就有很多很神仙的性质/应用

但对我来说就是一种优秀的点双缩点写法吧..

代码：APIO2018 铁人两项

#include<bits/stdc++.h>
#define LL long long 
using namespace std;
inline int read()
{
    int now = 0,fa = 1;char ch = getchar();
    for(;!isdigit(ch);ch = getchar())if(ch == '-') fa = -fa;
    for(;isdigit(ch);ch = getchar())now = 10 * now + ch - '0';
    return now * fa;
}
const int maxn = 100010;
int n,m;
struct Graph
{
    int first[maxn],to[maxn << 1],nx[maxn],cnt;
    inline void add(int u,int v)
    {
        to[++cnt] = v;
        nx[cnt] = first[u];
        first[u] = cnt;
    }
}G,Tree;
LL val[maxn],ans;int tot;
LL sum = 0;
int dfn[maxn],low[maxn],stk[maxn],ins[maxn],top,DFN,isc[maxn];
inline void Tarjan(int u,int fa)
{
    sum++; 
    dfn[u] = low[u] = ++DFN;stk[++top] = u;ins[u] = 1;
    val[u] = -1;
    for(int i=G.first[u];i;i=G.nx[i])
    {
        int targ = G.to[i];
        if(targ == fa)continue;
        if(!ins[targ])
        {
            Tarjan(targ,u);
            low[u] = min(low[u],low[targ]);
            if(low[targ] >= dfn[u])
            {
                tot++;
                val[tot] = 1;
                Tree.add(u,tot);
                do
                {
                    Tree.add(tot,stk[top]);
                    val[tot]++;
                }while(targ != stk[top--]);
            }
        }
        else low[u] = min(low[u],dfn[targ]);
    }
}
inline int dfs(int x,int type)
{
    LL size = type;
    for(int i=Tree.first[x];i;i=Tree.nx[i])
    {
        LL now = dfs(Tree.to[i],!type);
        ans += now * (sum - now) * val[x];
        size += now;
    }
    ans += (sum - size) * size * val[x];
    if(type) ans += (sum - 1ll) * val[x];
    return size;
}
int main()
{
    tot = n = read(),m = read();
    for(int i=1;i<=m;i++)
    {
        int u = read(),v = read();
        G.add(u,v);G.add(v,u);
    }
    for(int i=1;i<=n;i++)
        if(!ins[i])
        {
            sum = 0;
            Tarjan(i,0);
            dfs(i,1);
        }
    printf("%lld",ans);
}