We study the maximum induced matching problem on a graph g. Induced matchings correspond to independent sets in L2(g), the square of the line graph of g. The problem is NP-complete on bipartite graphs. In this work, we show that for a number of graph families with forbidden vertex orderings, almost all forbidden patterns on three vertices are preserved when taking the square of the line graph. These orderings can be computed in linear time in the size of the input graph. In particular, given a graph class G characterized by a vertex ordering, and a graph g = (V, E) in G with a corresponding vertex ordering σof V , one can produce (in linear time in the size of g) an ordering on the vertices of L2(g), that shows that L2(g) in G - for a number of graph classes G - without computing the line graph or the square of the line graph of g. These results generalize and unify previous ones on showing closure under L2(.) for various graph families. Furthermore, these orderings on L2(g) can be exploited algorithmically to compute a maximum induced matching on G faster. We illustrate this latter fact in the second half of the paper where we focus on cocomparability graphs, a large graph class that includes interval, permutation, trapezoid graphs, and co-graphs, and we present the first O(mn) time algorithm to compute a maximum weighted induced matching on cocomparability graphs; an improvement from the best known O(n4) time algorithm for the unweighted case.