On the strong chromatic index and maximum induced matching of   tree-cographs and permutation graphs
  We show that there exist linear-time algorithms that compute the strong chromatic index and a maximum induced matching of tree-cographs when the decomposition tree is a part of the input. We also show that there exists an efficient algorithm for the strong chromatic index of permutation graphs. 
Ton Kloks and Chin-Ting Ung and Yue-Li Wang
cs.DS