The planar multiterminal cut problem Original Research Article

Let G = (V, E) be a graph with positive edge weights and let Vʹ⊆V. The min V1-cut problem is to find a minimum weight set Eʹ⊆E such that no two nodes of Vʹ occur in the same component of Gʹ = (V, EEʹ). Our main results are two new structural theorems for optimal solutions to the min Vʹ-cut problem when G is planar. The first theorem establishes for the first time a close connection between the planar min Vʹ-cut problem and the well-known “Gomory-Hu” cut collections. The second theorem establishes a connection between the planar min Vʹ-cut problem and a particular matroid. Each theorem results in a simple algorithm for the planar min Vʹ-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.