Relation between k and k-1 edge connected graph Is every k connected graph is k-1 connected or the reverse? I always get confused. Can someone explain with the help of an example. 
 A: The answer is that it depends on your definition, and if you provide the definition you are using I can be more specific.


*

*The definition on mathworld is that a graph is $k$-connected if
    for every set of $k-1$ vertices, removing those vertices doesn't
    disconnect it. In this case, if a graph is $k$-connected then it is
    also $\ell$-connected for all $\ell<k$. To see this, notice that if
    removing many vertices cannot disconnect the graph, then you have no
    hope of disconnecting the graph by removing fewer vertices.

*The definition on wikipedia is that a graph is $k$-connected if
        $k$ is the smallest positive integer such that deleting $k$ vertices
        can disconnect the graph. Because of the use of the word "smallest,"
        a $k$-connected graph is neither $(k-1)$-connected nor
        $(k+1)$-connected. Under this definition, a clique on $m$ vertices is considered to be $(m-1)$-connected

*Another definition I have seen is that a graph is $k$-connected if
        there is a set of $k$ vertices such that removing them disconnects
        the graph. Under this definition, a graph that is
        $k$-connected is also $\ell$-connected for all $\ell > k$ since we
        can choose the $\ell$ vertices by first picking the $k$ vertices
        known to disconnect the graph, and then picking the remaining
        $\ell-k$ vertices arbitrarily.
Definition $1$ and $3$ come at the problem from "different sides" in some sense, and definition $2$ is equivalent to requiring that both definition $1$ and $3$ hold.
