Depth-first search in directed planar graphs, revisited

We present an algorithm for constructing a depth-first search tree in planar digraphs; the algorithm can be implemented in the complexity class \(\text{ AC}^1(\text{ UL }\cap \text{ co-UL})\), which is contained in \(\text{ AC}^2\). Prior to this (for more than a quarter-century), the fastest uniform deterministic parallel algorithm for this problem had a runtime of \(O(\log ^{10}n)\) (corresponding to the complexity class \(\text{ AC}^{10}\subseteq \text{ NC}^{11}\)). We also consider the problem of computing depth-first search trees in other classes of graphs and obtain additional new upper bounds.