vertex cover

The set of all vertices is a vertex cover.

Vertex cover is a special case of Hitting Set.

The endpoints of any maximal matching form a vertex cover.

A set is independent if and only if its complement is a vertex cover.

The complete bipartite graph has a minimum vertex cover of size .

It is minimum if there is no other vertex cover with fewer vertices.

This means that vertex cover is fixed-parameter tractable with the size of the solution as the parameter.

Then is a minimum vertex cover.

Since no smaller set of vertices could cover every edge in M, it must be a minimum vertex cover.

Irit Dinur contributed to the vertex cover problem.

The complement of A forms a vertex cover in G with the same cardinality as this matching.

One can find a factor-2 approximation by repeatedly taking both endpoints of an edge into the vertex cover, then removing them from the graph.

The "coverings" in the title of this paper refer to the vertex cover problem, not to bipartite double covers.

Furthermore, for any , vertex cover does not have kernels with edges unless .

On the other hand, the related problem of finding a smallest vertex cover is an NP-hard problem.

Similarly, the minimum vertex cover can be found as the complement of one of the maximal independent sets.

For instance, for the vertex cover problem, the parameter can be the number of vertices in the cover.

Moreover, if the unique games conjecture is true then minimum vertex cover cannot be approximated within any constant factor better than 2.

For example, the vertex cover problem and traveling salesman problem with triangle inequality each have simple 2-approximation algorithms.

In the vertex cover example, kernelization algorithms are known that produce kernels with at most vertices.

Here is the size of the largest matching and is the size of the smallest vertex cover.

If the problem is stated as a decision problem, it is called the vertex cover problem:

A vertex cover in a graph is a set of vertices that touches every edge in the graph.

Every vertex cover of size must contain since otherwise too many of its neighbors would have to be picked to cover the incident edges.

Thus, an optimal vertex cover for the original graph may be formed from a cover of the reduced problem by adding back to the cover.