# The Complexity of Planarity Testing

Conference paper

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## Abstract

We clarify the computational complexity of planarity testing, by showing that planarity testing is hard for **L**, and lies in **SL**. This nearly settles the question, since it is widely conjectured that **L** = **SL** [25]. The upper bound of **SL** matches the lower bound of **L** in the context of (nonuniform) circuit complexity, since **L**/poly is equal to **SL**/poly.

Similarly, we show that a planar embedding, when one exists, can be found in **FL** ^{SL}.

Previously, these problems were known to reside in the complexity class **AC** ^{1}, via a *O*(log *n*) time CRCW PRAM algorithm [22], although planarity checking for degree-three graphs had been shown to be in **SL** [23, 20].

## Keywords

Span Tree Planarity Testing Constraint Graph Fundamental Cycle Euler Tour
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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