Disjoint homotopic paths and trees in a planar graph

Abstract

In this paper we describe a polynomial-time algorithm for the following problem:given: a planar graph G embedded in ℝ2, a subset {I1, ..., Ip} of the faces of G, and paths C1, ..., Ck in G, with endpoints on the boundary of I1 ∪ ... ∪Ip; find: pairwise disjoint simple paths P1, ..., Pk in G so that, for each i=1, ..., k, Pi is homotopic to Ci in the space ℝ2\(I1 ∪ ... ∪Ip). Moreover, we prove a theorem characterizing the existence of a solution to this problem. Finally, we extend the algorithm to disjoint homotopic trees. As a corollary we derive that, for each fixed p, there exists a polynormial-time algorithm for the problem:given: a planar graph G embedded in ℝ2 and pairwise disjoint sets W1, ..., Wk of vertices, which can be covered by the boundaries of at most p faces of G;find: pairwise vertex-disjoint subtrees T1, ..., Tk of G where Ti (i=1, ..., k). © 1991 Springer-Verlag New York Inc.