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I have an undirected, unweighted graph representing a network. I have a starting node and an end one. My 'network' is reliable if there is no node such that without that node s and t are not reachable i.e. no node is necessary and all nodes have at least one redundant path.

how can I formally verify this? thank you

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What do you mean by "formally verify"? What do you mean that "all nodes have at least one redundant path"? –  Igor Rivin Sep 9 '12 at 18:22
    
As stated by Igor, you want biconnectivity. A naive way to check this is to check, for each vertex $v$, that $G-v$ is connected. –  Andrew D. King Sep 10 '12 at 20:25

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up vote 3 down vote accepted

If you mean that the network remains connected upon removing any one node, the magic words are "2-vertex-connected graph", or "biconnected graph". An algorithm for determining biconnectivity is described here, though I am sure there are plenty of other places.

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Btw, the OP has stated (without proof) a special case of Menger's theorem. –  Felix Goldberg Sep 9 '12 at 20:55

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