Abstract
For an integer l ⩾ 2 , the l-connectivity κ l ( 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 κ l ( G ) ⩾ k . A graph G is called minimally ( k , l ) -connected if κ l ( G ) ⩾ k but ∀ e ∈ E ( G ) , κ l ( 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.
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