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. In this paper, we present a structural characterization for minimally (2,l)-connected graphs and classify extremal results. These extend former results by Dirac (1967) [6], and Plummer (1968) [14] on minimally (2,2)-connected graphs.