# The Complexity of Planarity Testing

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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].

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