Bipartite graph
Note that to distinguish subject to a bipartite graph is a directed graph or undirected graph. For bipartite graph can be determined directly, and there needs to be demolished or point to a directed graph without constituting the corresponding bipartite graph (doubled although the number of nodes) and then using the formula (general formula for the last will divide \ (2 \) , while the original (non-directed graph or digraph) is the section points bipartite graph nodes \ (/ \) \ (2 \) ).
Minimum Vertex Cover: with a minimum point, to cover all sides. I.e., all sides at least one point of the endpoint has been used.
1, FIG bipartite Minimum Vertex Cover \ (= \) bipartite graph the maximum number of matches.
2, without covering the minimum point in FIG \ (= \) (the undirected bipartite graph nodes FIG \ (- \) of the non-maximum number of matches to FIG bipartite graph) \ (/ \) \ (2 \) .
Shortest path covered: with a minimum number of paths, to cover all the points.
1, there is a minimum to FIG disjoint path coverage: covering all points of the path with the least and only once all points on all paths.
Its value: number of points to FIG Section \ (- \) the maximum number of matches to the bipartite graph of FIG.
2, minimum path may intersect to FIG cover: cover all points of the path with the least, does not limit the number of times at each point appearing on all paths.
This question has finished in the construction of FIG. FIG performed transitive closure, then two points can reach all of A-> B are as a directed edge. And then on this new map has to be split points into bipartite graph, and then follow disjoint paths to write a cover.
Its value: number of points to FIG Section \ (- \) the maximum number of matches to the bipartite graph of FIG.
3, no coverage to FIG minimum path: For the content of this block, only on the network formula: undirected graph nodes \ (- \) Second FIG largest number of matched points \ (/ \) \ (2 \) . However, this formula is easy to verify is wrong.
For example, three points, \ (1-2-3 \) , the maximum number of the non-bipartite matching the view of FIG split \ (2 \) , and the minimum number of paths undirected FIG apparently \ (1 \ ) , it does not mean the formula \ (3 \) \ (- \) \ (2 \) \ (/ \) \ (2 \) .
Smallest side covered: with minimal side, to cover all the points. (Note the difference between the path)
1, bipartite graph Minimum edge covering: its value: original bipartite graph nodes (nodes and both sides) \ (- \) picture maximum number bipartite graph matching.
2, no minimum edge covering the drawing: it is :( corresponding bipartite graph nodes \ (- \) the maximum number of matches its corresponding bipartite graph) \ (/ \) \ (2 \) \ (= \ ) picture undirected graph nodes \ (- \) Second Panel largest number of matched \ (/ \) \ (2 \) .
Maximum independent set: up point selected pairwise no edge is connected between these points
FIG undirected maximum independent set: its value: no picture points to FIG Section \ (- \) the maximum number of matches bipartite graph corresponding \ (/ \) \ (2 \) .
Maximum group is independently: selected from the plurality of nodes on the original, both these points twenty-two edge connector. I.e. original maximum complete subgraph (up to vertex)
The maximum number of nodes independent groups \ (= \) maximum independent set up of FIG.