How to change undirected graph to point/edge double connectivity, how to turn a forward graph into a strong connected graph

Convert undirected graphs to point-to-double-connected graphs

Definition: Point-double connect refers to a path with at least two points that are not duplicated between any two

There are two kinds of cases, one is connected graph, the other is non-connected graph

① Connectivity Diagram

First, find all the points in the graph-the two connected components, and then indent the branch into a single point, because the inside of the two-connected component is definitely not considered.

It is only necessary to consider the two connected components and the other nodes outside, how to add the edge to form the point-double connected component.

　　　　

The node 2,3,4,5 constitutes a point-a double connected branch, which we indent to get the following figure

　　　　

As you can see, the graph after the indentation is a tree (because the rings are shrunk to a point)

Then the problem is converted to, if a tree becomes a point-a double-connected graph.

This special structure of the tree, as long as the leaf node between the edge, then you can make any two points there are two points do not repeat the path.

If there is only one leaf node, then just one edge to the root node.

If there are two leaf nodes, just one edge between the two leaf nodes.

If there are three leaf node a,b,c, then as long as a single edge between A, B, b,c a side can be

So the leaf node has cnt_leaf, so just Add (cnt_leaf+1)/2 side

② Non-connected graphs

For a non-connected graph, each connected component becomes a point-to-double connected component, then the method is the same as above.

Then there is only one problem if you make two points-a double-connected component into one.

Similarly, the point-to-double connected component as a point, then two points to become a point-double connectivity, then just add 2 edges on the line

　　

To turn undirected graphs into edges-double-connected graphs

Definition: There are at least two edges of the path that are not duplicated at any two points.

Also divided into two cases, one is connected graph, a non-connected graph

① Connectivity Diagram

Ditto

② Non-connected graphs

Ditto

Turn a forward graph into a strong connected diagram

Definition: A forward graph with any two points that can reach each other is called a strong connected graph.

　　

① Connectivity Diagram

Similarly, find all the strongly connected components and then indent to a point, then count the number of points (recorded as cntout) for the new graph after the indentation, and the number of points with a degree of 0 (recorded as Cntin)

So the number of edges to add is max (Cntout,cntin)

Why is this?? Because, if a point is in the degree of 0, then the point is not reached, if a point of the degree of 0, then the point to other points is not reached.

To resolve this situation, this is resolved only if a u-->v edge is connected between points with a degree of 0 (set to u) and a point with an entry degree of 0.

Continuous edge, as long as a point problem is not resolved to the edge, so is to take max between the two

② Non-connected graphs

For each connected branch, the above method becomes the strongly connected component.

As for the two connected components, a single side that you point to me, and a side that I point to you.

How to change undirected graph to point/edge double connectivity, how to turn a forward graph into a strong connected graph