"Collection" Graph theory related principles continue to update

Original: http://blog.csdn.net/leolin_/article/details/7199688

Minimum Edge overlay = maximum Independent set = | v| -Maximum number of matches this is done on the original and the binary chart.
The minimum path cover and the minimum side cover are different, the graph is not required to be a binary graph, but is required to be n x n the direction of the graph, can not have the ring, and then according to the original structure of the binary map, the construction method is to divide the point into two, for example, I is divided into I1 and i2 and if I and This makes up a two-part diagram

Then the minimum path overrides = N-m,n is the number of points of the original, and M is the maximum match for the new binary graph . Proving is also particularly simple, according to the definition of the minimum path coverage requires that the same point can only belong to a path, that is, the path can not be open fork, if in the binary chart selected two has a common point of the side so reaction in the original image is the path has a fork, so the two-part chart selected edge must be no public intersection, This is the conversion to the maximum match.

Summary:

"maximum independent number of graphs": a K-top is selected from the V-Vertex, which makes the K-top distinct from each other. So the largest k is the maximum independent number of the graph.

"maximal Group of Graphs": the K-Top is selected from the V-Vertex, which makes the K-top form a complete picture, that is, any two top of the sub-graph has a direct edge.

"Minimum Path overlay (the original image is not necessarily a binary graph, but must be a forward graph, split construction dichotomy)": find some paths in the diagram that cover all the vertices in the graph, and that any vertex has and only one path associated with it. Minimum path override = | v| -Maximum number of matches

Minimum Edge Overlay (Original is a binary chart): Find some edges in the diagram to cover all the vertices in the graph, and any one vertex has and only one edge associated with it. Minimum edge overlay = maximum Independent set = | v| -Maximum number of matches

Minimum vertex overlay: make each edge associated with at least one of the points with the fewest points (the points at the left and right sides of the collection).

PS: The original two-point matching when the collation of these figures, their relationship is a lot. such as: Minimum number of overlays + maximum Independent number = Vertex count. Although they are all NP-on. But there are good algorithms for special graphs, such as:

In a binary graph, the minimum number of overlays equals the maximum number of matches, and the maximum independent number equals the number of vertices minus the minimum number of overlays (= maximum match), so you can use Hungary to find the maximum independent number and so on.

A. Point cover set: A point set of the graph G, so that all edges in the graph have at least a point in the set.


B. Point Independent set: a point set of the graph G, so that any two points in the set are not adjacent to the original.


C. Minimum point cover Set: Point overlay set with least number of points in no direction graph G


D. Maximum Point independent set: No direction graph G, the point of the most points independent set


E. Minimum point weight coverage set: A point-weighted sum of points in a non-weighted graph


F. Maximum point weight independent set: The point right and the largest point independent set in the truth with the point right without the graph

Properties:

Maximum group = maximum independent set of complement graphs

Minimum side overlay = two min. max Independent set = | v| -Minimum Path coverage

Minimum path override = | v| -Maximum number of matches

minimum vertex overwrite = maximum number of matches

minimum vertex overlay + maximum Independent number = | v|

Minimum cut = minimum point weight cover set = Point right and-Maximum point weight independent set

Most powerful closed figure http://www.cnblogs.com/wuyiqi/archive/2012/03/12/2391960.html