Unavoidable doubly connected large graphs

A connected graph is double connected if its complement is also connected The following Ramsey type theorem is proved in this paper There exists a function h n de ned on the set of integers exceeding three such that every double connected graph on at least h n vertices must contain as an induced subgraph a special double connected graph which is either one of the following graphs or the complement of one of the following graphs Pn a path on n vertices K n the graph obtained from K n by subdividing an edge once K nne the graph obtained from K n by deleting an edge K n the graph obtained from K n by adding an edge between the two degree n vertices x and x and a pendent edge at each xi Two applications of this result are also discussed in the paper