Minimum vertex coverage of a binary graph maximum independent set maximum group

The minimum vertex overlay of a binary graph

Definition: If a point is selected, it is equivalent to overwriting all edges with which it is an endpoint. The minimum vertex overlay is to select the fewest points to cover all the edges.

Method: The minimum vertex overlay is equal to the maximum match of the two-point graph.

We use a binary graph to construct the minimum vertex overlay.

For the above dichotomy, the vertices are divided into two sets, and the X collection contains the 1,2,3,4,y collection containing the 5,6,7,8,9. If we have now found a maximum match m, that is, the above red line is marked by the m={(1,7), (2,5), (4,8)}. We define the following: (1) define 1, 2, 4, 5, 7, 8 as points that have already been matched, other points as mismatched points, (2) define (4,8), (1,7), (2,5) as matched edges, and other edges as unmatched.

Below we find the unmatched points from the Y collection, which are the 6 and 9 of the above markers. Every time we pick an unmatched point from the right, from that point on, do an unmatched edge---matching edge--match edge->......-> match Edge (note the end of the match edge), and mark the point used to get:

Where the purple side is the edge we just painted, where the marked points are 2, 4, 5, 6, 8, 9. The two routes are: (1) 9->4->8->2->5 (2) 6->2->5. Both roads are mismatched edges, matching edges, and unmatched edge->......-> matching edges. (Note that if a right-side unmatched point has multiple edges, then there are multiple paths starting at that point, with 6 and 9 having only one edge, so there is only one path from each point).

Now we will select the left marker point 2, 4 and the right unlabeled point 7 to make up the set S, that s is a minimum vertex cover set, that is, the S collection can cover all the edges. The following proves:

(1) | S|=m, that is, the minimum vertex overlay equals two The maximum match: the points on the left are all vertices that match the edges, because the point to the left is the maximum matching edge when we construct the path, and the point to the right is the vertex with the largest matching edge for the two-minute plot. And the left side of the tag plus having an edge unlabeled is exactly the number of maximum matches.

(2) s can cover all sides. All the edges can be divided into the following three cases: A, left point mark, right end mark, these edges must be covered by the point of the 2,4;b, such as the above, the right endpoint is not marked, the edges must be unmarked on the right side of the point, such as the above 7;c, left point unlabeled, right endpoint tag.

Here we prove that the side of C does not exist at all: if C is the edge of the largest match, because the right endpoint cannot be the starting point of a path (since our starting point is from the unmatched points in the Y set), then the right end of the mark can only be in the construction from the left side, this is the left end must be marked, C is then converted to A; If C is an unmatched edge, then the left endpoint must be a matching point, then the right end of C must be the starting point of a path, so the left end of C will also become the second point of a path, and C becomes a. So the C side certainly does not exist.

(3) S is the smallest set of vertices: because the maximum match is M, and | S|=m, so if there are fewer points in s, then even the M-matched edges cannot be overwritten.

The maximal independent set of the dichotomy graph

Definition: Selecting vertices so that these vertices are 22 nonadjacent, the set of these points is called a stand-alone set. Finding a stand-alone set with the highest number of vertices is called the maximum independent set.

Method: Maximum Independent set = all vertices-minimum vertex overlay

In the above figure, the minimum vertex overlay = 3, that is, the 2,4,7 constitutes the minimum vertex overlay, then the other points 6 constitute the largest independent set. and other points cannot be connected. Assuming that the other points are connected, the edge must not be 2,4,7 covered, and the 2,4,7 is the smallest vertex overlay contradiction. So there must be no edge between the other points. And 2,4,7 is the smallest vertex cover, so the so-called minimum is no longer small, so our independent set is the largest.

The largest group of the two-part graphs

Definition: For a general diagram, a regiment is a set of vertices, and the sub-graph induced by the set of vertices is a complete graph, simply put, is to select some vertices, these vertices 22 have edges. The largest group is the largest set of vertices chosen. For a binary plot, we default to all points on the left, with edges between all the vertices on the right. So, actually, we're going to find a vertex subset X on the left, and a subset of vertices y on the right, so that each vertex in X has an edge between each vertex in Y.

Method: Maximal independent set of the maximum group = complement graph of the binary graph.

The definition of a complement is: for the left point X and the right point y in the binary graph, if there is an edge between x and Y, then there is no in the complement graph, otherwise.

This method is well understood because the maximum independent set is 22 nonadjacent, so the complement of the largest independent set is 22 adjacent.

Minimum vertex coverage of a binary graph maximum independent set maximum group