Planarity, Determinants, Permanents, and (Unique) Matchings
ACM Transactions on Computation Theory (TOCT)
Limiting negations in bounded treewidth and upward planar circuits
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
Balancing bounded treewidth circuits
CSR'10 Proceedings of the 5th international conference on Computer Science: theory and Applications
Classification of planar upward embedding
GD'11 Proceedings of the 19th international conference on Graph Drawing
A generalization of spira's theorem and circuits with small segregators or separators
SOFSEM'12 Proceedings of the 38th international conference on Current Trends in Theory and Practice of Computer Science
The duals of upward planar graphs on cylinders
WG'12 Proceedings of the 38th international conference on Graph-Theoretic Concepts in Computer Science
On the curve complexity of upward planar drawings
CATS '12 Proceedings of the Eighteenth Computing: The Australasian Theory Symposium - Volume 128
Upward planar drawings on the standing and the rolling cylinders
Computational Geometry: Theory and Applications
Balancing Bounded Treewidth Circuits
Theory of Computing Systems
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The P-complete Circuit Value Problem CVP, when restricted to monotone planar circuits MPCVP, is known to be in NC3, and for the special case of upward stratified circuits, it is known to be in LogDCFL. In this paper we re-examine the complexity of MPCVP, with special attention to circuits with cylindrical embeddings. We characterize cylindricality, which is stronger than planarity but strictly generalizes upward planarity, and make the characterization partially constructive. We use this construction, and four key reduction lemmas, to obtain several improvements. We show that stratified cylindrical monotone circuits can be evaluated in LogDCFL, and arbitrary cylindrical monotone circuits can be evaluated in AC1(LogDCFL), while monotone circuits with one-input-face planar embeddings can be evaluated in LogCFL. For monotone circuits with focused embeddings, we show an upper bound of AC1(LogDCFL). We re-examine the NC3 algorithm for general MPCVP, and note that it is in AC1(LogCFL) = SAC2. Finally, we consider extensions beyond MPCVP. We show that monotone circuits with toroidal embeddings can, given such an embedding, be evaluated in NC. Also, special kinds of arbitrary genus circuits can also be evaluated in NC. We also show that planar non-monotone circuits with polylogarithmic negation-height can be evaluated in NC.