Find the number of undirected graphs on $\displaystyle n$ vertices such that, whenever there is a path $\displaystyle a_1 \leftrightarrow a_2 \leftrightarrow ... \leftrightarrow a_k$ joining the vertices $\displaystyle a_1,...,a_k$, then there is a complete graph on $\displaystyle a_1,...,a_k$.