Weighted undirected graph partitioning

Question

Given an undirected cyclic planar graph G(V,E) with vertex weights W(V), a fixed plane embedding E(G) and two nodes s and t, I need to find a partitioning of G that divides it into two connected components S(G) and T(G) with s being in S(G) and t being in T(G). Vertices s and t both belong to the external face in the embedding E(G).

I wish to have the partitions well balanced - they should have nearly equal sums of vertex weights.

Any ideas for a good algorithm please?