The DLT conference series provides a forum to present ongoing advances in formal languages and automata. Its scope has always been very general, ranging from fundamental to application-related aspects. The edition of 2022 followed this trend as can be seen from the proceedings of DLT 2022 [1] published with Lecture Notes in Computer Science. This Special Issue contains extended versions of selected contributions presented at DLT 2022. These contributions were nominated by the Program Committee by voting, based on the conference reviews and the impression that the papers have given when being presented at the conference. The issue also includes papers based on the invited talks by Joel Day and Jarkko Kari.

The paper studies Σ 0 n -computable families (n ⩾ 2) and their numberings. It is proved that any non-trivial Σ 0 n -computable family has a complete with respect to any of its elements Σ 0 n -computable non-principal numbering. It is established that if a Σ 0 n -computable family is not principal, then any of its Σ 0 n -computable numberings has a minimal cover and, if the family is infinite, is incomparable with one of its minimal Σ 0 n -computable numberings. It is also shown that for any Σ 0 ncomputable numbering ν of a Σ 0 n -computable non-principal family there exists its Σ 0 n -computable numbering that is incomparable with ν. If a non-trivial Σ 0 ncomputable family contains the least and greatest elements under inclusion, then for any of its Σ 0 n -computable non-principal non-least numberings ν there exists a Σ 0 n -computable numbering of the family incomparable with ν. In particular, this is true for the family of all Σ 0 n -sets and for the families consisting of two inclusion-comparable Σ 0 n -sets (semilattices of the Σ 0 n -computable numberings of such families are isomorphic to the semilattice of m-degrees of Σ 0 n -sets).

The rational index ρL of a language L is an integer function, where ρL(n) is the maximum length of the shortest string in L ∩ R, over all regular languages R recognized by n-state nondeterministic finite automata (NFA). This paper investigates the rational index of languages defined by grammars with bounded tree dimension, and shows that it is polynomial in n. More precisely, it is proved that for a context-free grammar with tree dimension bounded by d, its rational index is at most O(n 2d ), and that this estimation is asymptotically tight, as there exists a grammar with rational index Θ(n 2d ). For a multi-component grammar of rank k and with tree dimension bounded by d, the rational index is bounded by O(n 2kd ), and there exists a grammar with rational index Ω(n 2kd ).

The original version of this article was revised to update the presentation of Table 1 . The correct table is also shown below. The original article has been corrected. Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The s-Club problem asks whether a given undirected graph G contains a vertex set S of size at least k such that G[S], the subgraph of G induced by S, has diameter at most s. We consider variants of s-Club where one additionally demands that each vertex of G[S] is contained in at least ell triangles in G[S], that each edge of G[S] is contained in at least ell triangles in G[S], or that S contains a given set W of seed vertices. We show that in general these variants are W[1]-hard when parameterized by the solution size k, making them significantly harder than the unconstrained s-Club problem. On the positive side, we obtain some FPT algorithms for the case when ell =1 and for the case when G[W], the graph induced by the set of seed vertices, is a clique.

located in Ekaterinburg, Russian Federation. This symposium traditionally covers a broad range of topics in Theoretical Computer Science, ranging from fundamental to application-related. The edition of 2020 published its proceedings with Lecture Notes in Computer Science [1], as it was the case with its precursors in this conference series. This Special Issue contains extended versions of selected contributions that were previously presented at CSR 2020. These contribution were nominated by the Program Committee of CSR 2020 by voting, based on the conference reviews and on the impression that the papers have given when being presented at the conference. This selection also shows the breadth of the research topics that have been present at CSR 2020, which is actually quite typical for this conference series, ranging from formal languages to parameterized and other forms of complexity, clearly also including algorithmic questions. More specifically, the following papers have been chosen: -"On the parameterized complexity of the Expected Coverage problem" by Fedor V. Fomin and Vijayaragunathan Ramamoorthi; -"On the decision tree complexity of threshold functions" by Anastasiya Chistopolskaya and Vladimir V. Podolskii; -"Kernelization of Arc Disjoint Cycle Packing in α-bounded digraphs" by Abhishek Sahu and Saket Saurabh;

In k-Digraph Coloring we are given a digraph and are asked to partition its vertices into at most k sets, so that each set induces a DAG. This well-known problem is NP-hard, as it generalizes (undirected) k-Coloring, but becomes trivial if the input digraph is acyclic. This poses the natural parameterized complexity question of what happens when the input is “almost” acyclic. In this paper we study this question using parameters that measure the input’s distance to acyclicity in either the directed or the undirected sense. In the directed sense perhaps the most natural notion of distance to acyclicity is directed feedback vertex set. It is already known that, for all k ≥ 2, k-Digraph Coloring is NP-hard on digraphs of directed feedback vertex set of size at most k + 4. We strengthen this result to show that, for all k ≥ 2, k-Digraph Coloring is already NP-hard for directed feedback vertex set of size exactly k. This immediately provides a dichotomy, as k-Digraph Coloring is trivial if directed feedback vertex set has size at most k − 1. Refining our reduction we obtain three further consequences: (i) 2-Digraph Coloring is NP-hard for oriented graphs of directed feedback vertex set at most 3; (ii) for all k ≥ 2, k-Digraph Coloring is NP-hard for graphs of feedback arc set of size at most k2; interestingly, this leads to a second dichotomy, as we show that the problem is FPT by k if feedback arc set has size at most k2 − 1; (iii) k-Digraph Coloring is NP-hard for graphs of directed feedback vertex k, even if the maximum degree Δ is at most 4k − 1; we show that this is also almost tight, as the problem becomes FPT for digraphs of directed feedback vertex set of size k and Δ ≤ 4k − 3. Since these results imply that the problem is also NP-hard on graphs of bounded directed treewidth, we then consider parameters that measure the distance from acyclicity of the underlying graph. On the positive side, we show that k-Digraph Coloring admits an FPT algorithm parameterized by treewidth, whose parameter dependence is (tw!)ktw. Since this is considerably worse than the ktw dependence of (undirected) k-Coloring, we pose the question of whether the tw! factor can be eliminated. Our main contribution in this part is to settle this question in the negative and show that our algorithm is essentially optimal, even for the much more restricted parameter treedepth and for k = 2. Specifically, we show that an FPT algorithm solving 2-Digraph Coloring with dependence tdo(td) would contradict the ETH. Then, we consider the class of tournaments. It is known that deciding whether a tournament is 2-colorable is NP-complete. We present an algorithm that decides if we can 2-color a tournament in \(O^{*}({\sqrt [3]{6}}^{n})\) time. Finally, we explain how this algorithm can be modified to decide if a tournament is k-colorable.