# Cut point and Bridge formwork

Algorithm:

dfn [u] is the first visited u in dfs traversal.

low [u] is the depth of the ancestor with the earliest access time that the dfs u can reach.

First, we can get the dfn and low of each point by traversing the graph once.

Consider all points u and its son nodes s1,s2 in the dfs tree sk.

(1) If low [si] > = DFN [u] is satisfied for a certain si, then u is a cut point;

(2) If low [si] > DFN [u] is satisfied for a certain si, then the edge (u,si) is a bridge.

For the root node of the search tree, we need to make a special judgment: if the root has only one child node, then the root must not be the cut point.

Template:

```const int MAX_V = 1000;
const int MAX_E = 1000000;

int vis[MAX_V]; //Node v's current access status, 0 indicates no access, 1 indicates in the stack, and 2 indicates that it has been accessed
int dfn[MAX_V]; //Depth at which node v is accessed
int low[MAX_V]; //The depth of the earliest ancestor that node v can reach
bool cut[MAX_V];
bool bridge[MAX_V][MAX_V];
//Cur is the condition of cutting point: ① cur is the root and has more than one son; ② cur is not the root and has one son, V makes low [v] > = DFN [cur];
//(cur, i) is the condition of bridge: low [i] > DFN [cur];
void cut_bridge(int cur, int father, int dep, int n) //vertex: 0~n-1
{
vis[cur] = 1;
dfn[cur] = dep;
low[cur] = dep;
int children = 0;
for(int i = 0; i < n; i++)
{
if(edge[cur][i]) //Node i connected to the current node
{
if(i != father && vis[i] == 1) //i in the current stack, it shows that there is a ring in the diagram, and the low value of cur is updated with the depth of i;
{
if(dfn[i] < low[cur]) low[cur] = dfn[i]; //Update the depth of the ancestor with the earliest access time that the node cur can reach to the depth when the node i is accessed;
}
if(vis[i] == 0) //I has not been visited, recursively accesses node i, and updates the low value of cur with the earliest reachable ancestor of I;
{
cut_bridge(i, cur, dep+1, n);
children ++;

if(low[i] < low[cur]) low[cur] = low[i];
if((father == -1 && children > 1) || (father == -1 && low[i] >= dfn[cur]) //Judgement cut point
cut[cur] = true;
if(low[i] > dfn[cur]) bridge[cur][i] = bridge[i][cur] = true;             //Judgement Bridge
}
}
}
vis[cur] = 2;
}
//For each connected block, take a point X and call cut ˊ bridge (x, - 1,0, n), where n is the number of points.```

Posted by nawhaley2265 on Thu, 09 Jan 2020 11:35:21 -0800