Polynomial Algorithms for (integral) Maximum Two-flows in Vertex Edge-capacitated Planar Graphs

In this paper we study the maximum two-flow problem in vertexand edge-capacitated undirected STI-planar graphs, that is, planar graphs where the vertices of each terminal pair arc 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 ST>-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 STz-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. Kep~nds: Planar graphs; Vertex and edge capacitated; Two-flow; Integral; Algorithms