-neighborhood closures for graphs

For any positive integer k and any ( 2 + k − n ) -connected graph of order n , we define, following Bondy and Chvatàl, the k -neighborhood closure N C k ( G ) as the graph obtained from G by recursively joining pairs of nonadjacent vertices a , b satisfying the condition | N ( a ) ∪ N ( b ) | + δ a b + ε a b ≥ k , where δ a b = min { d ( x ) | a , b ∉ N ( x ) ∪ { x } } and ε a b is a well defined binary variable. For many properties P of G , there exists a suitable k (depending on P and n ) such that N C k ( G ) has property P if and only if G does.