NP-completeness of the Planar Separator Problems

For a given graph G, the Separator Problem asks whether a vertex or edge set of small cardinality (or weight) exists whose removal partitions G into two disjoint graphs of approximately equal sizes. Called the Vertex Separator Problem when the removed set is a vertex set, and the Edge Separator Problem when it is an edge set, both problems are NP-complete for general unweighted graphs [6]. Despite the significance of planar graphs, it has not been known whether the Planar Separator Problem, which considers a planar graph and a threshold as an input, is NP-complete or not. In this paper, we prove that the Vertex Separator Problem is in fact NP-complete when G is a vertex weighted planar graph. The Edge Separator Problem will be shown NP-complete when G is a vertex and edge weighted planar graph. In addition, we consider how to treat the constant α ∈ R + of the α-Separator Problem that partitions G into two disjoint graphs of size at most (1− α) |V (G)|. The α-Separator Problem is not NP-complete for all real numbers α ∈ (0, 1/2], because it would imply uncountably many Non Deterministic Turing Machines. We will present a general scheme for treating a constant in computer arithmetic, by introducing the notion of real numbers comparable with rationals in polynomial time. This approach allows us to prove NP-completeness for each such real number α. Article Type Communicated by Submitted Revised regular paper M. Fürer November 2001 October 2005 J. Fukuyama, The Planar Separator Problems, JGAA, 10(2) 317–328 (2006)318