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

Given $k$ pairs of terminals ${(s{1}, t{1}), \ldots, (s{k}, t{k})}$ in a graph $G$, the min-sum $k$ vertex-disjoint paths problem is to find a collection ${Q{1}, Q{2}, \ldots, Q{k}}$ of vertex-disjoint paths with minimum total length, where $Q{i}$ 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 polynomial-time algorithm for $k \le 3$ 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. We make progress toward Colin de Verdiere and Schrijver's open question by giving an $O(kn^5)$ time algorithm for undirected planar graphs when ${(s{1}, t{1}), \ldots, (s{k}, t{k})}$ are in counter-clockwise order on a common face.