The Planar Multiterminal Cut Problem

Let G = (V, E) be a graph with positive edge weights and let V’ c V. The min VI-cut prohlenl is to find a minimum weight set E’ E E such that no two nodes of V’ occur in the same component of G’ = (V, E\E’). Our main results are two new structural theorems for optimal solutions to the min V-cut problem when G is planar. The first theorem establishes for the first time a close connection between the planar min VI-cut problem and the well-known “GomoryP Hu” cut collections. The second theorem establishes a connection between the planar min V’-cut problem and a particular matroid. Each theorem results in a simple algorithm for the planar min V’-cut problem. The first algorithm is based upon the most efficient previous algorithm for this problem (due to Dahlhaus et al.) and achieves a lower time complexity.