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k-Edge-Connected Graph

A graph is -edge-connected if there does not exist a set of edges whose removal disconnects the graph (Skiena 1990, p. 177). The maximum edge connectivity of a given graph is the smallest degree of any node, since deleting these edges disconnects the graph. Complete bipartite graphs have maximum edge connectivity.

-edge-connectedness graph checking is implemented in the Wolfram Language as KEdgeConnectedGraphQ[g, k].

The following table gives the numbers of -edge-connected graphs for -node graphs.

 OEIS , 2, ... 0 A000719 0, 1, 2, 5, 13, 44, 191, ... 1 A052446 0, 1, 1, 3, 10, 52, 351, ... 2 A052447 0, 0, 1, 2, 8, 41, 352, ... 3 A052448 0, 0, 0, 1, 2, 15, 121, ... 4 0, 0, 0, 0, 1, 3, 25, ... 5 0, 0, 0, 0, 0, 1, 3, ... 6 0, 0, 0, 0, 0, 0, 1, ...

Cyclic Edge Connectivity, Edge Connectivity, Edge Cut, Graph Bridge, k-Connected Graph

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References

Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 45, 1994.Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990.Sloane, N. J. A. Sequences A000719/M1452, A052446, A052447, and A052448 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

k-Edge-Connected Graph

Cite this as:

Weisstein, Eric W. "k-Edge-Connected Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/k-Edge-ConnectedGraph.html