  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 $s_i$-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 $k \le 3$ 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.
