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 pairwise vertex-disjoint paths connecting the pairs ( s i , t i ) in G , with minimal total length, in O ( kn log n ) time.