Cut (graph theory)

Cut (graph theory)

In graph theory, a cut is a partition of the vertices of a graph into two disjoint subsets. The cut-set of the cut is the set of edges whose end points are in different subsets of the partition. Edges are said to be crossing the cut if they are in its cut-set.

In an unweighted undirected graph, the size or weight of a cut is the number of edges crossing the cut. In a weighted graph, the same term is defined by the sum of the weights of the edges crossing the cut.

In a flow network, an s-t cut is a cut that requires the source and the sink to be in different subsets, and its cut-set only consists of edges going from the source's side to the sink's side. The capacity of an s-t cut is defined by the sum of capacity of each edge in the cut-set.

The cut of a graph can sometimes refer to its cut-set instead of the partition.



A cut C = (S,T) is a partition V of a graph G = (V,E).
An s-t cut C = (S,T) of a network N = (V,E) is a cut of N such that s\in S and t \in T, where s and t are the source and the sink of N respectively.
The cut-set of a cut C = (S,T) is the set \{(u,v)\in E | u\in S, v \in T\}.
The size of a cut C = (S,T) is the number of edges in the cut-set. If the edges are weighted, the value of the cut is the sum of the weights.

Minimum cut

A minimum cut.

A cut is minimum if the size of the cut is not larger than the size of any other cut. The illustration on the right shows a minimum cut: the size of this cut is 2, and there is no cut of size 1 because the graph is bridgeless.

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.

Maximum cut

A maximum cut.

A cut is maximum if the size of the cut is not smaller than the size of any other cut. The illustration on the right shows a maximum cut: the size of the cut is equal to 5, and there is no cut of size |E| because the graph is not bipartite (there is an odd cycle).

In general, finding a maximum cut is computationally hard. The max-cut problem is one of Karp's 21 NP-complete problems. The max cut problem is also APX-hard, meaning that there is no polynomial-time approximation scheme for it unless P = NP.

Note that min-cut and max-cut are not dual problems in the linear programming sense, even though one gets from one problem to other by changing min to max in the objective function. The max-flow problem is the dual of the min-cut problem.

Sparsest cut

The Sparsest cut problem is to bipartition the vertices so as to minimize the ratio of the number of edges across the cut divided by the number of vertices in the smaller half of the partition. This objective function favors solutions that are both sparse (few edges crossing the cut) and balanced (close to a bisection). The problem is known to be NP-Hard, and the best known algorithm is an O(\sqrt{\log n}) approximation due to Arora, Rao & Vazirani (2009).

See also


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Graph theory — In mathematics and computer science, graph theory is the study of graphs : mathematical structures used to model pairwise relations between objects from a certain collection. A graph in this context refers to a collection of vertices or nodes and …   Wikipedia

  • Connectivity (graph theory) — In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) which need to be removed to disconnect the remaining nodes from each other[1]. It is… …   Wikipedia

  • Minor (graph theory) — In graph theory, an undirected graph H is called a minor of the graph G if H is isomorphic to a graph that can be obtained by zero or more edge contractions on a subgraph of G. The theory of graph minors began with Wagner s theorem that a graph… …   Wikipedia

  • Bridge (graph theory) — A graph with 6 bridges (highlighted in red) An undirected connected graph with no cut …   Wikipedia

  • Vertex (graph theory) — For other uses, see Vertex (disambiguation). A graph with 6 vertices and 7 edges where the vertex number 6 on the far left is a leaf vertex or a pendant vertex In graph theory, a vertex (plural vertices) or node is the fundamental unit out of… …   Wikipedia

  • Glossary of graph theory — Graph theory is a growing area in mathematical research, and has a large specialized vocabulary. Some authors use the same word with different meanings. Some authors use different words to mean the same thing. This page attempts to keep up with… …   Wikipedia

  • List of graph theory topics — This is a list of graph theory topics, by Wikipedia page. See glossary of graph theory for basic terminology Contents 1 Examples and types of graphs 2 Graph coloring 3 Paths and cycles 4 …   Wikipedia

  • König's theorem (graph theory) — In the mathematical area of graph theory, König s theorem describes an equivalence between the maximum matching problem and the minimum vertex cover problem in bipartite graphs. Setting A graph is bipartite if its vertices can be partitioned into …   Wikipedia

  • Transportation network (graph theory) — A transportation network is a type of directed, weighted graph or network.Transportation networks are used to model the flow of commodity, information, or traffic (see transport network). Definitions A transportation network G is a graph with… …   Wikipedia

  • Cut — may refer to: The act of cutting, the separation of an object into two through acutely directed force Contents 1 Mathematics 2 Computing 3 …   Wikipedia