Contractible elements in k‐connected graphs not containing some specified graphs

In 15 , Thomassen proved that any triangle‐free k‐connected graph has a contractible edge. Starting with this result, there are several results concerning the existence of contractible elements in k‐connected graphs which do not contain specified subgraphs. These results extend Thomassen's result, cf., 2 , 3 , 9–13 . In particular, Kawarabayashi 12 proved that any k‐connected graph without K  ‐4 subgraphs contains either a contractible edge or a contractible triangle.