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In graph theory, a minimum cut or min-cut of a graph is a cut (a partition of the vertices of a graph into two disjoint subsets) that is minimal in some metric. Variations of the minimum cut problem consider weighted graphs, directed graphs, terminals, and partitioning the vertices into more than two sets.
The max-flow min-cut theorem proves that the maximum network flow and the sum of the cut-edge weights of any minimum cut that separates the source and the sink are equal. There are polynomial-time methods to solve the min-cut problem, notably the Edmonds–Karp algorithm. [2]
For an unweighted graph, the minimum cut would simply be the cut with the least edges. For a weighted graph, the sum of all edges' weight on the cut determines whether it is a minimum cut. In practice, the minimum cut problem is always discussed with the maximum flow problem, to explore the maximum capacity of a network, since the minimum cut ...
A graph and two of its cuts. The dotted line in red is a cut with three crossing edges. The dashed line in green is a min-cut of this graph, crossing only two edges. In computer science and graph theory, Karger's algorithm is a randomized algorithm to compute a minimum cut of a connected graph. It was invented by David Karger and first ...
A variant of the problem asks for a minimum weight k-cut where the output partitions have pre-specified sizes. This problem variant is approximable to within a factor of 3 for any fixed k if one restricts the graph to a metric space, meaning a complete graph that satisfies the triangle inequality. [7]
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.
In the undirected edge-disjoint paths problem, we are given an undirected graph G = (V, E) and two vertices s and t, and we have to find the maximum number of edge-disjoint s-t paths in G. Menger's theorem states that the maximum number of edge-disjoint s-t paths in an undirected graph is equal to the minimum number of edges in an s-t cut-set.
Graph homomorphism problem [3]: GT52 Graph partition into subgraphs of specific types (triangles, isomorphic subgraphs, Hamiltonian subgraphs, forests, perfect matchings) are known NP-complete. Partition into cliques is the same problem as coloring the complement of the given graph. A related problem is to find a partition that is optimal terms ...