Pair of vertex disjoint cycles in a directed graph What is the fastest known deterministic algorithm that can recognize directed graphs with a pair of vertex disjoint cycles? I know graphs with min outdegree three always have such a pair (Thomassen'83), but even so I cannot find an efficient algorithm in the general case. Does anyone know a reference for this?
According to Grohe and Grüber "Parameterized approximability of the disjoint cycle problem" (ICALP 2007) there is an algorithm for finding $k$ vertex-disjoint cycles in a digraph, in time $n^{f(k)}$ for some function $f$ (polynomial for fixed $k$ but not FPT) in the section 5 of Reed, Robertson, Seymour and Thomas, "Packing directed circuits" (Combinatorica 1996) (which in turn uses theorem 3 of "The directed subgraph hemeomorphism problem" of Fortune, Hopcroft, and Wyllie.)