Constant-width circuit characterization of NC^1

Develop a characterization of NC^1 in terms of constant-width circuits that is not provided by crossing number, genus, or thickness.

Background

The paper investigates whether NC^1, which lies between ACC^0 and the larger class characterized by unrestricted constant-width circuits, can be characterized using constant-width circuits with graph structures intermediate between planar and unrestricted graphs. The results show that polylogarithmic genus and crossing number still characterize ACC^0, while thickness two already suffices for the larger class NC^1. Consequently, these prominent graph-theoretic candidates do not yield the desired characterization, and the authors explicitly designate finding an alternative characterization as the principal open problem.