Parallel algorithms for maximal acyclic sets

Given a graph G=(V, E), the classical spanning forest problem of G can be viewed as the problem of finding a maximal subset F of E inducing an acyclic subgraph. Although it is well known that this problem has efficient NC algorithms, its vertex counterpart, i.e., the problem of finding a maximal subset U of V inducing an acyclic subgraph, has not been shown to be in NC (or even in RNC) and is not believed to be parallelizable in general. We present NC algorithms for solving the latter problem for three special cases. The first algorithm solves the problem for planar graphs in O(log³ n) time using O(n) processors on an EREW PRAM. The second algorithm solves the problem for K_3,3-free graphs in O(log⁴ n) time using O(n) processors on an EREW PRAM. The third algorithm solves the problem for graphs without long induced paths in poly-logarithmic time using O(n²³⁷⁶) processors on an EREW PRAM