Abstract
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 � 2 R
Highlights
The Separator Problem questions whether a vertex or edge set of small cardinality exists in a given graph G whose removal partitions G into two disjoint graphs of approximately equal sizes
Let α ∈
We have introduced the notion of a real number comparable with rationals in polynomial time, and have shown that the Planar α-Vertex Separator Problem for a vertex weighted planar graph G is NP-complete for every fixed real number α ∈
Summary
The Separator Problem questions whether a vertex or edge set of small cardinality (or weight) exists in a given graph G whose removal partitions G into two disjoint graphs of approximately equal sizes. It is called the Vertex Separator Problem when the removed set is a vertex set, and the Edge Separator Problem when it is an edge set. An algorithm of poly-logarithmic approximation ratio has been discovered for the Graph Bisection Problem [5], which is a variant of the Edge Separator Problem that partitions G into two disjoint subgraphs of the exactly equal size.
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