ββ-neighborhood closures for graphs

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