  We first design an $\mathcal{O}(n^2)$ solution for finding a maximum induced
matching in permutation graphs given their permutation models, based on a
dynamic programming algorithm with the aid of the sweep line technique. With
the support of the disjoint-set data structure, we improve the complexity to
$\mathcal{O}(m + n)$. Consequently, we extend this result to give an
$\mathcal{O}(m + n)$ algorithm for the same problem in trapezoid graphs. By
combining our algorithms with the current best graph identification algorithms,
we can solve the MIM problem in permutation and trapezoid graphs in linear and
$\mathcal{O}(n^2)$ time, respectively. Our results are far better than the best
known $\mathcal{O}(mn)$ algorithm for the maximum induced matching problem in
both graph classes, which was proposed by Habib et al.
