The following idea is easier to comprehend practically, unlike the connectivity of G which is not always easy to determine.

A graph G is k-connected if 

     We can characterize the non-trivial cases (for k> 0): 1-connected graphs are connected graphs such that n ≥ 2, and 2-connected graphs are connected graphs with no cut vertex and such that n ≥ 3.

Notes

1)If k^/≥ k then k^/-connected implies k-connected.

2) In a graph, the blocks which have at least three vertices may be seen as 2-connected components of this graph (components in the sense of induced subgraph maximal according to the property considered). Going back over and comparing the two ideas which we have just defined,  the connectivity of G and the property of k-connectivity, we can state (integer k is assumed to be ≥ 0):

1) A graph G verifies κ(G)= k if and only if n ≥ k +1, G − A is connected for any A ⊆ X such that |A| <k and there exists A ⊆ X such that |A| = k and G − A is disconnected or reduced to one vertex.

2) A graph G is k-connected if and only if n ≥ k + 1 and G − A is connected for any A ⊆ X such that |A| <k.

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Graph Theory  and Applications ,Jean-Claude Fournier, WILEY, page(62-63)

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