寒假2019培训：双连通分量（点双+边双）

定义：

若一个无向连通图不存在割点，则称它为“点双连通图”。若一个无向连通图不存在割边，则称它为“边双连通图”。
无向图的极大点双连通子图称为“点双连通分量”，简记为“v-DCC”。无向连通图的极大边双连通子图被称为“边双连通分量”，简记为“e-DCC”。二者统称为“双连通分量”，简记为“DCC”。

边双联通分量

求法：

核心概念： 没有割边
割边只会把图分成两部分，对图中的点没有影响。

用红色临摹出来的，便是割边（请看我的另一个博客了解割边）
Tarjan学习+割点+割边+强连通分量

删除他们

上面就有三个边双了~~

算法：只需求出无向图中所有的割边，把割边都删除后，无向图会分成若干个连通块，每一个连通块就是一个“边双连通分量”。
具体实现：一般先用Tarjan算法求出所有的桥，然后再对整个无向图执行一次dfs遍历（遍历的过程不访问割边），划分出每个连通块即可。

贴上代码~

void dfs(int x){color[x] = tot;for(int i = linkk[x];i;i = e[i].n)if(!color[e[i].y] && !e[i].flag) //e[i].flag表示i为割边dfs(e[i].y);
}
void make_node(){t = 0;memset(linkk,0,sizeof(linkk));for(int i = 1;i <= m;++i){int j = i * 2; int x = e[j].x , y = e[j].y;if(color[x] != color[y]) insert( color[x] , color[y] ) ;}return;
}
// 主程序for(int i = 1;i <= n;++i)if(!color[i]) ++tot , dfs(i);make_node();

上一道模板题吧

bzoj 1718: [Usaco2006 Jan] Redundant Paths 分离的路径

题目描述
In order to get from one of the F (1 <= F <= 5,000) grazing fields (which are numbered 1…F) to another field, Bessie and the rest of the herd are forced to cross near the Tree of Rotten Apples. The cows are now tired of often being forced to take a particular path and want to build some new paths so that they will always have a choice of at least two separate routes between any pair of fields. They currently have at least one route between each pair of fields and want to have at least two. Of course, they can only travel on Official Paths when they move from one field to another. Given a description of the current set of R (F-1 <= R <= 10,000) paths that each connect exactly two different fields, determine the minimum number of new paths (each of which connects exactly two fields) that must be built so that there are at least two separate routes between any pair of fields. Routes are considered separate if they use none of the same paths, even if they visit the same intermediate field along the way. There might already be more than one paths between the same pair of fields, and you may also build a new path that connects the same fields as some other path.

为了从F(1≤F≤5000)个草场中的一个走到另一个，贝茜和她的同伴们有时不得不路过一些她们讨厌的可怕的树．奶牛们已经厌倦了被迫走某一条路，所以她们想建一些新路，使每一对草场之间都会至少有两条相互分离的路径，这样她们就有多一些选择． 每对草场之间已经有至少一条路径．给出所有R(F-1≤R≤10000)条双向路的描述，每条路连接了两个不同的草场，请计算最少的新建道路的数量, 路径由若干道路首尾相连而成．两条路径相互分离，是指两条路径没有一条重合的道路．但是，两条分离的路径上可以有一些相同的草场． 对于同一对草场之间，可能已经有两条不同的道路，你也可以在它们之间再建一条道路，作为另一条不同的道路．

输入格式
Line 1: Two space-separated integers: F and R * Lines 2…R+1: Each line contains two space-separated integers which are the fields at the endpoints of some path.

第1行输入F和R，接下来R行，每行输入两个整数，表示两个草场，它们之间有一条道路．

输出格式
Line 1: A single integer that is the number of new paths that must be built.

最少的需要新建的道路数．

样例数据
input

7 7
1 2
2 3
3 4
2 5
4 5
5 6
5 7
output

2
样例解释：

Building new paths from 1 to 6 and from 4 to 7 satisfies the conditions.
1 2 3
±–±--+
: | |
: | |
6 ±–±--+ 4
/ 5 :
/ :
/ :
7 + - - - -
Check some of the routes:
1 - 2: 1 -> 2 and 1 -> 6 -> 5 -> 2
1 - 4: 1 -> 2 -> 3 -> 4 an