In the Directed Detour problem one is given a digraph G and a pair of vertices s and t, and the task is to decide whether there is a directed simple path from s to t in G whose length is larger than dist G (s, t). The more general parameterized variant, Directed Long Detour, asks for a simple s-to-t path of length at least dist G (s, t) + k, for a given parameter k. Surprisingly, it is still unknown whether Directed Detour is polynomial-time solvable on general digraphs. However, for planar digraphs, Wu and Wang [Networks, '15] proposed an O(n 3 )-time algorithm for Directed Detour, while Fomin et al. [STACS 2022] gave a 2 O(k) •n O(1) -time fpt algorithm for Directed Long Detour. The algorithm of Wu and Wang relies on a nontrivial analysis of how short detours may look like in a plane embedding, while the algorithm of Fomin et al. is based on a reduction to the 3-Disjoint Paths problem on planar digraphs. This latter problem is solvable in polynomial time using the algebraic machinery of Schrijver [SIAM J. Comp., '94], but the degree of the obtained polynomial factor is huge.In this paper we propose two simple algorithms: we show how to solve, in planar digraphs, Directed Detour in time O(n 2 ) and Directed Long Detour in time 2 O(k) • n 4 log n. In both cases, the idea is to reduce to the 2-Disjoint Paths problem in a planar digraph, and to observe that the obtained instances of this problem have a certain topological structure that makes them amenable to a direct greedy strategy.

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Let ϕ be a sentence of CMSO 2 (monadic second-order logic with quantification over edge subsets and counting modular predicates) over the signature of graphs. We present a dynamic data structure that for a given graph G that is updated by edge insertions and edge deletions, maintains whether ϕ is satisfied in G. The data structure is required to correctly report the outcome only when the feedback vertex number of G does not exceed a fixed constant k, otherwise it reports that the feedback vertex number is too large. With this assumption, we guarantee amortized update time O ϕ,k (log n).By combining this result with a classic theorem of Erdős and Pósa, we give a fully dynamic data structure that maintains whether a graph contains a packing of k vertexdisjoint cycles with amortized update time O k (log n). Our data structure also works in a larger generality of relational structures over binary signatures.