Shortest Vertex-Disjoint Two-Face Paths In Planar Graphs

Let G be a directed planar graph of complexity n, each arc having a nonnegative length. Let s and t be two distinct faces of G; let s(1), . . ., s(k) be vertices incident with s; let t(1), . . . ,t(k) be vertices incident with t. We give an algorithm to compute k pair-wise vertex-disjoint paths connecting the pairs (s(i), t(i)) in G, with minimal total length, in O(kn log n) time.