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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 family of all cut sets of an undirected graph is known as the cut space of the graph. It forms a vector space over the two-element finite field of arithmetic modulo two, with the symmetric difference of two cut sets as the vector addition operation, and is the orthogonal complement of the cycle space.
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 ...
The minimum cut found in all phases will be the minimum weighted cut of the graph. A cut is a partition of the vertices of a graph into two non-empty, disjoint subsets. A minimum cut is a cut for which the size or weight of the cut is not larger than the size of any other cut. For an unweighted graph, the minimum cut would simply be the cut ...
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.
Minimum k-cut; Minimum k-spanning tree; Minor testing (checking whether an input graph contains an input graph as a minor); the same holds with topological minors; Steiner tree, or Minimum spanning tree for a subset of the vertices of a graph. [2] (The minimum spanning tree for an entire graph is solvable in polynomial time.)
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In graph theory and combinatorial optimization, a closure of a directed graph is a set of vertices C, such that no edges leave C. The closure problem is the task of finding the maximum-weight or minimum-weight closure in a vertex-weighted directed graph. [1] [2] It may be solved in polynomial time using a reduction to the maximum flow problem.
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