Towards single face shortest vertex-disjoint paths in undirected planar graphs

Abstract

Given k pairs of terminals {(s1, t1),..., (sk, tk)} in a graph G, the min-sum k vertex-disjoint paths problem is to find a collection {Q1,Q2,...,Qk} of vertex-disjoint paths with minimum total length, where Qi is an si-to-ti path between si and ti. We consider the problem in planar graphs, where little is known about computational tractability, even in restricted cases. Kobayashi and Sommer propose a polynomialtime 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(kn5) time algorithm for undirected planar graphs when {(s1, t1),..., (sk, tk)} are in counter-clockwise order on a common face.