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The Complexity of Planarity Testing
, 2000
"... 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) circ ..."
Abstract

Cited by 23 (7 self)
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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 degreethree graphs had been shown to be in SL [23, 20].
Planarity, Determinants, Permanents, and (Unique) Matchings
"... Abstract. We explore the restrictiveness of planarity on the complexity of computing the determinant and the permanent, and show that both problems remain as hard as in the general case, i.e. GapL and #P complete. On the other hand, both bipartite planarity and bimodal planarity bring the complexity ..."
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Abstract. We explore the restrictiveness of planarity on the complexity of computing the determinant and the permanent, and show that both problems remain as hard as in the general case, i.e. GapL and #P complete. On the other hand, both bipartite planarity and bimodal planarity bring the complexity of permanents down (but no further) to that of determinants. The permanent or the determinant modulo 2 is complete for ⊕L, and we show that parity of paths in a layered grid graph (which is bimodal planar) is also complete for this class. We also relate the complexity of grid graph reachability to that of testing existence/uniqueness of a perfect matching in a planar bipartite graph. 1