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.
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
来源：https://stackoverflow.com/questions/10115114/determining-if-a-graph-is-k-vertex-connected