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I'm looking to come up with an polynomial time algorithm that takes input in the form of a graph, G, and an integer, K, and determines whether G is K-vertex connected. I'm thinking that this would likely utilize Depth First Search. I can see how it could be none with a none-polynomial solution, i.e. just deleting K random vertices, running DFS to check for connectivity, and then doing it again with a different group of vertices. A run time of ~O(n^K) is a little much though, and it is apparently possible to bring this down to polynomial time. Any idea what I'm missing here? I imagine it has something to do with the non-tree vertices that we get after running a DFS, but I'm not totally sure what I'm looking for? Thanks in advance!

Edit: To be clear, i am not looking to determine the connectivity of the graph. Rather, a number, k, is given on input and I am looking to check if the graph is k connected. It will not produce an answer that gives the connectivity of the graph, just a yes or no.

user1257768
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  • The fastest algorithm to determine the vertex connectivity of a graph is due to Henzinger, Rao, and Gabow (Computing Vertex Connectivity: New Bounds from Old Techniques) and runs in time O(min{k^3 + n, kn}m), where n is the number of vertices, m is the number of edges, and k is the connectivity. This is a comment not an answer because the article is behind a paywall and I don't feel like summarizing. – oldboy Apr 11 '12 at 23:39
  • Ah yes, in case it wasn't clear above I am _not_ looking to determine the vertex connectivity of an input graph (I'm aware that that is not doable in polynomial time), but rather just to check if a graph is k-connected. So I would get a graph and, say, the number 4, and check whether or not the graph is 4 connected. Does this make sense? – user1257768 Apr 12 '12 at 13:07

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You can compute vertex-connectivity for an input graph in polynomial time, even when k is not fixed, see https://en.wikipedia.org/wiki/K-vertex-connected_graph#Computational_complexity

a3nm
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