Cut (graph theory)

A section referred to in graph theory, a set of edges of a graph, which is located between two sets of nodes, or between an amount and the remaining amount.

Particular importance is attached to sections related to networks. Sections can be defined and studied by networks but also independently.

Definition

For a non-directed graph, and a portion of the section is: defined as:

It thus covers all edges for which it holds that a terminal node is in the subset and the other in the amount of the other nodes. These edges are, so to speak "between" the two subsets of the nodes.

In directed graphs, there are different definitions of the term "cut", a widely used definition is:

Obviously applies here unlike undirected graph that can be.

Another way to define cuts in directed graphs, is to take the edge in first, regardless of their orientation, so that in turn would apply is. In this case, would be in the two subsets and disintegrate. Then holds that either or, it is called a directed cut, ie show either all directed edges out into the crowd into it or out of it.

Important phrases and statements

Connection and minimal cuts

If you were to remove all the edges of a cut from the graph, it would no longer give way between and, and the resulting graph would thus have at least one connected component more. Was the graph before removing the edges of the cut together hanging, he is subsequently no longer.

In this context, a cut is also referred to as the minimum section, when, after the removal of the edges of the cut exactly two connection components are produced from the graph. It can be shown that this is precisely the case when a set of nodes can be selected such that the current induced by their intersection does not contain subsets of edges forming an induced by another node set section. In short, a cut is minimal, if not already a subset of the section forms a section.

Disjoint paths and cuts

The mathematician Karl Menger found a connection between nodes and edge-disjoint paths and cuts. This set of Menger was generalized to the Max -Flow Min -Cut theorem later.

One considers a connected graph with two subsets of the nodes and. Between two nodes and examined at the number of edge-disjoint paths as well as the cardinality (ie number of edges ) of a section. Since all edge-disjoint paths from to have by the intersection ( because the removal of the edges of the cut destroyed so all paths from to) and, since the paths must be edge-disjoint, each time a different edge of the cut must be used, is obviously true that the cardinality of the cut must be at least as large as the number of edge-disjoint paths between and. Menger finally showed that the maximum number of edge-disjoint paths of a minimum separating edge set corresponds exactly.

This finding applies both in directed and in the undirected graph. It can be further transmitted by edge-disjoint paths for node-disjoint and then states that the maximum number knotendisjunkter paths between two nodes and the cardinality of a minimal separating vertex set corresponds.

This then implies the k-connectedness of a graph not only that you have to remove at least nodes to destroy the relationship, but also that it always must be a minimum node-disjoint paths between any two nodes of a graph.

Sections and groups

Even between cuts and circles, there are some relationships. So is the cardinality of the intersection of the edges of a section of a circle and, thus be straight. Imagine a circular edge in front, for which it holds that it is additionally also on the cut, so you have without loss of generality and his. If the circle would now start in and then use, then the considered edge is not the only one cutting edge of the circle and cut, as it is now in the subset and still have to use an odd number of edges between the circle and cut to get back in the subset switch back in which is located. Overall, it must therefore be an even number of edges.

In a to the dual graph cuts to circles and circles to be cuts.

If a directed graph is strongly connected, you can turn considered node sets. Then must apply to all possible sets that the cut is. According to the definition of directed cuts that is equivalent to the statement that there should be no directed cut. Because with a "real" choice of then could there be a directed cut, which must mean, by definition, that is, but this would contradict the previous statement.