Towards single face shortest vertex - disjoint paths in undirected 1 planar graphs

7 Given k pairs of terminals {(s1, t1), . . . , (sk, tk)} in a graph G, the min-sum k vertex-disjoint 8 paths problem is to find a collection {Q1, Q2, . . . , Qk} of vertex-disjoint paths with minimum 9 total length, where Qi is an si-to-ti path between si and ti. We consider the problem in 10 planar graphs, where little is known about computational tractability, even in restricted cases. 11 Kobayashi and Sommer propose a polynomial-time algorithm for k ≤ 3 in undirected planar 12 graphs assuming all terminals are adjacent to at most two faces. Colin de Verdière and Schrijver 13 give a polynomial-time algorithm when all the sources are on the boundary of one face and all 14 the sinks are on the boundary of another face and ask about the existence of a polynomial-time 15 algorithm provided all terminals are on a common face. 16 We make progress toward Colin de Verdière and Schrijver’s open question by giving an 17 O(kn) time algorithm for undirected planar graphs when {(s1, t1), . . . , (sk, tk)} are in counter18 clockwise order on a common face. 19