Geodesic
Edge-disjoint paths in permutation graphs
Discussiones Mathematicae. Graph Theory, Tome 15 (1995) no. 1, pp. 59-72

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In this paper we consider the following problem. Given an undirected graph G = (V,E) and vertices s₁,t₁;s₂,t₂, the problem is to determine whether or not G admits two edge-disjoint paths P₁ and P₂ connecting s₁ with t₁ and s₂ with t₂, respectively. We give a linear (O(|V|+|E|)) algorithm to solve this problem on a permutation graph.
Keywords: algorithm, bridge, connectivity, disjoint paths, permutation graph 
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Gopalakrishnan, C.; Pandu Rangan, C. Edge-disjoint paths in permutation graphs. Discussiones Mathematicae. Graph Theory, Tome 15 (1995) no. 1, pp. 59-72. http://geodesic.mathdoc.fr/item/DMGT_1995_15_1_a6/