Restricted edge connectivity and restricted connectivity of graphs

Let G = (V, E) be a connected graph. Let G = (V, E) be a connected graph. An edge set F ⊂ E is said to be a k-restricted edge cut, if G− F is disconnected and every component of G− F has at least k vertices. The k-restricted edge connectivity of G, denoted by λk(G), is the cardinality of a minimum k-restricted edge cut of G. A graph G is λk-connected, if G contains a k-restricted edge cut. A λk-connected graph G is called λk-optimal, if λk(G) = ξk(G), where ξk(G) = min{|[U, V − U ]| : U ⊂ V, |U | = k and G[U ] is connected}.An vertex set X is a k-restricted cut of G, if G −X is not connected and every component of G −X has at least k vertices. The k-restricted connectivity κk(G) (in short κk) of G, is the cardinality of a minimum k-restricted cut of G. A λk-connected graph G is said to be super-λk, if G is λk-optimal and every minimum k-restricted edge cut isolates a component with exactly k vertices. A κk-connected graph G is said to be super-κk, if κ3(G) = ξ3(G) and the deletion of each minimum k-restricted cut isolates a component with exactly k vertices. In this paper, we study the restricted edge connectivity and restricted connectivity of graphs, line graphs and a kind of transformation graphs. Key–Words: 3-Restricted edge connectivity; Super-λ3; Super-κ3