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| {{DISPLAYTITLE:''k''-edge-connected graph}}
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| In [[graph theory]], a graph is '''''k''-edge-connected''' if it remains [[connectivity (graph theory)|connected]] whenever fewer than ''k'' edges are removed.
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| ==Formal definition==
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| Let <math>G = (V, E)</math> be an arbitrary graph.
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| If subgraph <math>G' = (V, E \setminus X)</math> is connected for all <math>X \subseteq E</math> where <math>|X| < k</math>, then ''G'' is ''k''-edge-connected. Trivially, a graph that is ''k''-edge-connected is also (''k''−1)-edge-connected.
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| ==Relation to minimum vertex degree==
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| Minimum [[degree (graph theory)|vertex degree]] gives a trivial upper bound on edge-connectivity. That is, if a graph <math>G = (V, E)</math> is ''k''-edge-connected then it is necessary that ''k'' ≤ δ(''G''), where δ(''G'') is the minimum degree of any vertex ''v'' ∈ ''V''. Obviously, deleting all edges incident to a vertex, ''v'', would then disconnect ''v'' from the graph.
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| == Computational aspects ==
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| There is a polynomial-time algorithm to determine the largest ''k'' for which a graph ''G'' is ''k''-edge-connected. A simple algorithm would, for every pair ''(u,v)'', determine the [[Maximum flow problem|maximum flow]] from ''u'' to ''v'' with the capacity of all edges in ''G'' set to 1 for both directions. A graph is ''k''-edge-connected if and only if the maximum flow from ''u'' to ''v'' is at least ''k'' for any pair ''(u,v)'', so ''k'' is the least ''u-v''-flow among all ''(u,v)''.
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| If ''n'' is the number of vertices in the graph, this simple algorithm would perform <math>O(n^2)</math> iterations of the Maximum flow problem, which can be solved in <math>O(n^3)</math> time. Hence the complexity of the simple algorithm described above is <math>O(n^5)</math> in total.
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| An improved algorithm will solve the maximum flow problem for every pair ''(u,v)'' where ''u'' is arbitrarily fixed while ''v'' varies
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| over all vertices. This reduces the complexity to <math>O(n^4)</math> and is sound since, if a [[Cut_(graph_theory)|cut]] of capacity less than ''k'' exists,
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| it is bound to separate ''u'' from some other vertex. It can be further improved by [[Gabow's algorithm]] that runs in worst case <math>O(n^3)</math> time. <ref> Harold N. Gabow. A matroid approach to finding edge connectivity and packing arborescences. ''J. Comput. Syst. Sci.'', 50(2):259–273, 1995.</ref>
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| A related problem: finding the minimum ''k''-edge-connected subgraph of ''G'' (that is: select as few as possible edges in ''G'' that your selection is ''k''-edge-connected) is NP-hard for <math>k\geq 2</math>.<ref>M.R. Garey and D.S. Johnson. ''Computers and Intractability: a Guide to the Theory of NP-Completeness''. Freeman, San Francisco, CA, 1979.</ref>
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| == See also ==
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| * [[k-vertex-connected graph]]
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| * [[Connectivity (graph theory)]]
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| * [[Matching preclusion]]
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| * [[Menger's theorem]]
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| * [[Robbins theorem]]
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| ==References==
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| {{reflist}}
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| [[Category:Graph connectivity]]
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| [[Category:Graph families]]
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