Towards Single Face Shortest Vertex-Disjoint Paths in Undirected Planar Graphs

Abstract

Given k pairs of terminals {(s ₁, t ₁), …, (s _k , t _k )} in a graph G, the min-sum k vertex-disjoint paths problem is to find a collection {Q ₁, Q ₂, …, Q _k } of vertex-disjoint paths with minimum total length, where Q _i is an s _i -to-t _i path between s _i and t _i . We consider the problem in planar graphs, where little is known about computational tractability, even in restricted cases. Kobayashi and Sommer propose a polynomial-time algorithm for k ≤ 3 in undirected planar graphs assuming all terminals are adjacent to at most two faces. Colin de Verdière and Schrijver give a polynomial-time algorithm when all the sources are on the boundary of one face and all the sinks are on the boundary of another face and ask about the existence of a polynomial-time algorithm provided all terminals are on a common face.

We make progress toward Colin de Verdière and Schrijver’s open question by giving an O(kn ⁵) time algorithm for undirected planar graphs when {(s ₁, t ₁), …, (s _k , t _k )} are in counter-clockwise order on a common face.