K-edge-connected graph

The edge connectivity of a graph is an important concept in graph theory and a generalization of context. Clearly, the edge-connectivity is a measure of how hard it is to decompose a graph by deleting edges in two components. If the edge-connectivity large, so many edges must be deleted.

Definition

A simple graph is called k -fold edge-connected, if a maximum -element edge set and an empty set of nodes are in ( in multigraph can be edges according to their multiplicity remove multiple) no separator. Equivalent to this is that is contiguous for all Kantenmeengen of cardinality. As edge-connectivity number of a graph is defined as the largest so - fold edge-connected. Equivalent to this is that the edge-connectivity number is the least cardinality of a separator with an empty node-set.

Example

Consider as an example the house shown on the right of Nicholas. There is not 1 - edge Contiguous because there are no separators, which consist of only one edge. Equivalent to this is that there is no bridge. But if we consider, for example, now node 5, then the graph decomposes when deleting the two edges incident to node 5 in two related components: the node 5 itself and all other nodes. The house is therefore 1-fold and 2 -fold edges Contiguous and its edge-connectivity number. In this case, the edge-connectivity number so true accordance with the minimum degree of the graph.

Properties

  • If so applies, any node connection speed and the minimum degree of the graph.
  • If and only 2-fold edge-connected if no bridge.
  • Is exactly then fold edge-connected, if contains edge-disjoint paths between any two vertices. This statement is also known as the global version of Menger's theorem.

Related Tendings

The k- edge-connectivity for connection is a similar term, merely that only separators are considered with an empty edge set and an arbitrary set of nodes. K- connection thus gives a measure of how many nodes have to be removed from a graph, so that this is divided into different components. A similar term for edge-connectivity for directed graphs is the arc connection

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