In this paper we study the maximum two-flow problem in vertex-and edge-capacitated undirected ST2-planar graphs, that is, planar graphs where the vertices of each terminal pair are on the same face. For such graphs we provide an O(n) algorithm for finding a minimum two-cut and an O(n log n) algorithm for determining a maximum two-flow and show that the value of a maximum two-flow equals the value of a minimum two-cut. We further show that the flow obtained is half-integral and provide a characterization of edge and vertex capacitated ST2-planar graphs that guarantees a maximum two-flow that is integral. By a simple variation of our maximum two-flow algorithm we then develop, for ST2-planar graphs with vertex and edge capacities, an O(n log n) algorithm for determining an integral maximum two-flow of value not less than the value of a maximum two-flow minus one.
Granot, F., & Penn, M. (1996). Polynomial algorithms for (integral) maximum two-flows in vertex\edge-capacitated planar graphs. Discrete Applied Mathematics, 70(3), 267–283. https://doi.org/10.1016/0166-218X(95)00111-4