Generalizations of critical connectivity of graphs

Abstract We generalized the concepts of a pragment and an atom of a graph and show that these generalizations have properties similar to the common concepts. We prove that a contraction- critical, finite graph G has at least ∣ G ∣/3 triangles and that a finite graph G is 8-connected if every complete subgraph of G is contained in a smallest separating set of G . We study some further classes of graphs (almost critical graphs, C k -critical graphs) and discuss some applications.