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

Abstract

Given k pairs of terminals {(s1, t1), …, (s k , t k )} in a graph G, the min-sum k vertex-disjoint paths problem is to find a collection {Q1, Q2, …, 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 Verdiere 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.