Unavoidable double - connected large

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), deened 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: (1) P n , a path on n vertices; (2) K s 1;n , the graph obtained from K 1;n by subdividing an edge once; (3) K 2;n ne, the graph obtained from K 2;n by deleting an edge; (4) K + 2;n , the graph obtained from K 2;n by adding an edge between the two degree-n vertices x 1 and x 2 , and a pendent edge at each x i. Two applications of this result are also discussed in the paper.