Minimum path coverage and minimum edge coverage and related properties

"Minimum Path overlay"

First, the formula is given: the minimum path of the DAG spans the number of nodes in the =dag graph-the maximum number of matches in the corresponding binary graph.

In a PXP graph, the path overlay is to find a path in the diagram that overrides 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.

It can be concluded from the above: 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. For a path overlay, there are the following properties: 1, each vertex belongs to and belongs to only one path. 2. On the path except the end point, only one edge from each vertex points to another vertex on the path. So how to construct the corresponding sub-graph for a dag? For a vertex p in a DAG, there are two vertices p and P ' in the two-dimensional graph, corresponding to a forward-edged p->q in the DAG, and a non-directed edge of P-q ' in the binary graph. P belongs to the s set in two, and P ' belongs to T collection.

, corresponding to the left side of the DAG to construct the right side of the sub-graph, you can find a binary map of a maximum match m:1-3 ', 3-4 ', then the M two matching edges how to correspond to the path in the Dag edge? One edge in a binary graph corresponds to a forward edge in a dag, and 1-3 ' corresponds to a directed edge 1 in a DAG >3, so that 1 in the DAG will have a successor vertex (3 will be the only successor of 1, because a vertex in the binary graph is associated with at most one edge!), so 1 does not become the end vertex in a path in the DAG, and similarly, 3-4 ' corresponds to a dag where 3->4,3 does not become the end vertex. Then a total of 4 vertices in the original image, minus 2 has a successor vertex, there is no successor vertex, that is, the end of the DAG path vertex, and each end vertex exactly corresponds to a path in the DAG, the binary graph to find the maximum match m, is to find the corresponding DAG in the maximum number of non-path end vertices, So the number of vertices in the Dag-| M| is the minimum number of end vertices in the DAG, which is the minimum path coverage for the DAG. That is, the minimum path that is found in the overlay collection is 2, 1->3->4. "Minimum Edge overlay"

Edge Overlay set: In layman's sense, the so-called edge overlay set is that all vertices in g are adjacent vertices (edge overlay vertices) of an edge in e*, and an edge can only cover 2 vertices.

Note: There are few edges to "overwrite" all vertices in a non-graph, so the edge cover set has a minimum and minimum difference.

Minimal Edge overlay: If any true subset in the edge overlay e* is not an edge overlay set, then the e* is called a minimal edge overlay set.

Minimum Edge overlay: The minimum edge overlay is called the minimum edge overlay, and in layman's sense, it is the smallest set in the minimal edge overlay.

Application of the minimum edge overlay in a binary graph: minimum edge overlay = maximum Independent set = N-Maximum match , this is a property of the binary graph.

"Binary Graph related properties"

In a binary graph, the number of point overlays is the number of matches.

(1) The maximum matching number of the binary graph is equal to the minimum covering number, that is, the least point makes each edge at least one point associated with it, it is obvious that the maximum matching node can be directly taken.

(2) The independent number of the binary graph is equal to the vertex number minus the maximum number of matches, it is obvious that the points at both ends of the maximum match are removed from the vertex set this time the remaining points are independent sets, which is | v|-2*| M|, at the same time must be from each side of the matching edge

Take a point to join the independent set and maintain its independent set nature.

(3) The minimum path of the DAG overlay, each point is split after the maximum match, the result is n-m, the specific path of the time along the matching side to go, matching edge i→j ', J→k ', k→l ' .... form a forward path.

(4) Maximum number of matches = left match point + right unmatched point. Because on either side of the maximum matching set, if his left is unmarked and the right is marked, then we can find a new augmented path, so that each edge is at least

Be covered by a dot.

(5) Minimum side overlay = number of points in the graph-maximum number of matches = Maximum independent set.

Minimum path coverage and minimum edge coverage and related properties