The Complexity of Planarity Testing Eric Allender Meena Mahajan Conference paper First Online: 24 March 2000 DOI :
10.1007/3-540-46541-3_7

Part of the
Lecture Notes in Computer Science
book series (LNCS, volume 1770) Cite this paper as: Allender E., Mahajan M. (2000) The Complexity of Planarity Testing. In: Reichel H., Tison S. (eds) STACS 2000. STACS 2000. Lecture Notes in Computer Science, vol 1770. Springer, Berlin, Heidelberg 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 ].

Supported in part by NSF grant CCR-9734918.

Part of this work was done when this author was supported by the NSF grant CCR-9734918 on a visit to Rutgers University during summer 1999.

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Authors and Affiliations Eric Allender Meena Mahajan 1. Dept. of Computer Science Rutgers University Piscataway USA 2. The Institute of Mathematical Sciences Chennai India