Characterization of minimally (2, l)-connected graphs

For an integer l ≥ 2, the l-connectivity κl(G) of a graph G is defined to be the minimum number of vertices of G whose removal produces a disconnected graph with at least l components or a graph with fewer than l vertices. Let k ≥ 1, a graph G is called (k, l)-connected if κl(G) ≥ k. A graph G is called minimally (k, l)-connected if κl(G) ≥ k but ∀e ∈ E(G), κl(G − e) ≤ k − 1. We present a structural characterization for minimally (2, l)-connected graphs and classify extremal results. These extend former results by Dirac and Plummer on minimally (2,2)-connected graphs.