# Minimum path coverage, minimum point coverage, maximum independent point set

Source: Internet
Author: User

Node 1: Minimum Path overlay

In a PXP graph, the path overlay is to look for paths in the diagram that cover all the vertices in the graph, and that any vertex has and only one path associated with it; (if each path in these paths goes from its starting point to its end, it can go through each vertex of the diagram once and only once) , if the circuit is not considered in the diagram, then each path is a weakly connected subset.
From the above can be drawn:
1. A single vertex is a path;
2. If there is a path p1,p2,...... PK, where P1 is the starting point, PK is the end point, then in the overlay map, the vertex p1,p2,...... PK no longer exists with the other vertices have a forward edge.
The minimum path overlay is to find the smallest number of path bars, making it a path overlay of p.
The relationship between the path overlay and the binary map match:
Minimum path coverage =|p|-maximum match number;
The maximum matching number is calculated by dividing each vertex pi in P into two vertex pi ' and pi ', if there is a pi-to-PJ edge in P, then there is a non-direction link between pi ' and PJ ' in the binary graph P ', where Pi ' is the out side of pi in P, PJ ' is P PJ An entry edge;

For the formula: the minimum path overrides =|p|-maximum match number;

You can understand this:

If the match number is zero, then there is no forward edge in P, so there is obviously:

Minimum path coverage =|p|-maximum match number =|p|-0=|p|;

That is, the minimum path coverage number of P is |p|;

The number of path overlays is |p| when the matching edge is not in P ';

If you add a matching edge pi '-->pj ' in P ', then there is a PI connection to PJ's Edge in the path overlay of figure p, that is, the Pi and PJ are on a path, so the path cover number can be reduced by one;

So continue to increase the matching edge, each increase, the number of path coverage is reduced by one, until the matching edge can not continue to increase, the number of path coverage can no longer be reduced, there is a previous formula; but it's just talking. Each matching edge corresponds to a path in the path overlay to a link between two points of the forward edge The following is a description of each of the two vertices in a path overlay corresponding to a matching edge;

Similar to the previous, for each connection in the path overlay two vertices between each of the forward-edge pi--->pj, we can be in the matching graph to make a link to the pi ' and PJ ' side, it is clear that the figure is a matching diagram (which is easy to prove with contradiction, If the resulting graph is not a matching graph, then there must be such two-edge pi '---pj ' and pi '----PK ', (j!=k), then there are two edges PI-->PJ in the path overlay, pi--->pk, There is more than one path from the PI, which is contradictory to the path overlay diagram, and there is another case where pi '---pj ', pk '---PJ ', is similar to the case;

At this point, it shows that the matching edge and the path overlay between the two vertices connected to the edge of the one by one correspondence, then also shows that the previous formula is set up!

(Excerpt from: http://www.cppblog.com/SHFACM/archive/2009/02/05/73050.html)

The following questions are these topics, you can practice practiced hand:

Zoj 1364 | | POJ 1325

Zoj 1525 | | POJ 1422

Node 2: Minimum point overlay

In the binary chart, select the minimum number of points so that the points are associated with all edges (covering all edges), called the minimum point overlay.
Minimum point coverage = maximum number of matches

Proof: http://hi.baidu.com/keeponac/blog/item/1764bec86f820f8dc91768b7.html

Node 3: Maximum set of independent points

In a binary graph, select vertices so that no edges are connected between any of the two vertices in the selected point set to prove that the maximum number of independent vertex sets = total vertices-Maximum matches excerpted from: http://my.opera.com/IloveLunamaria/blog/ show.dml/810972 PS: From the above conclusion, the minimum coverage point set and the maximum independent point set complementary, that is, the minimum point coverage + Maximum Independent point set = total number of vertices similar, in the sub-graph with the point right, the minimum point weight coverage set + Maximum point right independent set = Total weight and

Minimum path coverage, minimum point coverage, maximum independent point set

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