Linear-Time Algorithms for Disjoint Two-Face Paths Problems in Planar Graphs

In this paper we present a linear-time algorithm for the vertex-disjoint Two-Face Paths Problem in planar graphs, i.e., the problem of finding k vertex-disjoint paths between pairs of terminals which lie on two face boundaries. The algorithm is based on the idea of finding rightmost paths with a certain property in planar graphs. Using this method, a linear-time algorithm for finding vertex-disjoint paths of a certain homotopy is derived. Moreover, the algorithm is modified to solve the more general linkage problem in linear time as well