Search results
Results from the WOW.Com Content Network
An example of a bipartite graph, with a maximum matching (blue) and minimum vertex cover (red) both of size six. In the mathematical area of graph theory, Kőnig's theorem, proved by Dénes Kőnig (), describes an equivalence between the maximum matching problem and the minimum vertex cover problem in bipartite graphs.
The minimum vertex cover problem is the optimization problem of finding a smallest vertex cover in a given graph. INSTANCE: Graph OUTPUT: Smallest number such that has a vertex cover of size . If the problem is stated as a decision problem, it is called the vertex cover problem:
The optimization problem of finding such a set is called the maximum independent set problem. It is a strongly NP-hard problem. [3] As such, it is unlikely that there exists an efficient algorithm for finding a maximum independent set of a graph. Every maximum independent set also is maximal, but the converse implication does not necessarily hold.
In particular, this shows that any maximal matching is a 2-approximation of a maximum matching and also a 2-approximation of a minimum maximal matching. This inequality is tight: for example, if G is a path with 3 edges and 4 vertices, the size of a minimum maximal matching is 1 and the size of a maximum matching is 2.
The vertex-connectivity statement of Menger's theorem is as follows: . Let G be a finite undirected graph and x and y two nonadjacent vertices. Then the size of the minimum vertex cut for x and y (the minimum number of vertices, distinct from x and y, whose removal disconnects x and y) is equal to the maximum number of pairwise internally disjoint paths from x to y.
The corresponding NP optimization problem of finding the size of a minimum feedback vertex set can be solved in time O(1.7347 n), where n is the number of vertices in the graph. [3] This algorithm actually computes a maximum induced forest, and when such a forest is obtained, its complement is a minimum feedback vertex set.
Any minimum feedback vertex set in the graph in Figure 1 has three vertices: for example, the three vertices A, C, and G. It is possible to construct examples in which the gap between the two quantities - the size of the largest collection of vertex-disjoint cycles, and the size of the smallest feedback vertex set - is arbitrarily large.
A minimum edge covering is an edge covering of smallest possible size. The edge covering number ρ(G) is the size of a minimum edge covering. The following figure shows examples of minimum edge coverings (again, the set C is marked with red). Note that the figure on the right is not only an edge cover but also a matching.