In the Multiterminal Cut problem we are given an edge-weighted graph and a subset of the vertices called terminals, and asked for a minimum weight set of edges that separates each terminal from all the others. When the number k of terminals is two, this is simply the mincut, max-flow problem, and can be solved in polynomial time. We show that the problem becomes NP-hard as soon as k = 3, but can be solved in polynomial time for planar graphs for any fixed k. The planar problem is NP-hard, however, if k is not fixed. We also describe a simple approximation algorithm for arbitrary graphs that is guaranteed to come within a factor of 2 − 2/ k of the optimal cut weight.

A module of an undirected graph G = V E is a set X of vertices that have the same set of neighbors in V \X. The modular decomposition is a unique decomposition of the vertices into nested modules. We give a practical algorithm with an O n + mα m n time bound and a variant with a linear time bound.

Absfrast. In the Multiway Cut problem we are given an edgaweighted graph and a subset of the vertices called termimds, and asked for a minimum weight set of edges that separates each terminal from all the others. when the number k of terminals is two, this is simply the min-cu~msx-flow problem, and can be solved in polynomial time. We show that the problem becomes NP-hsrd as sxrn as k = 3, but ctm be solved in polynomial time for planar graphs for any fixed k. The planar problem is NP-hsrd however, if k is not tixad. We also describe a simple approximation slgorithnt for arbkmry graphs that is guaranteed to come within a fsctor of 2-2/k of the optimal cut weight.

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