  We consider well-quasi-order for classes of permutation graphs which omit
both a path and a clique. Our principle result is that the class of permutation
graphs omitting $P_5$ and a clique of any size is well-quasi-ordered. This is
proved by giving a structural decomposition of the corresponding permutations.
We also exhibit three infinite antichains to show that the classes of
permutation graphs omitting $\{P_6,K_6\}$, $\{P_7,K_5\}$, and $\{P_8,K_4\}$ are
not well-quasi-ordered.
