Monadic stability generalizes many tameness notions from structural graph theory such as planarity, bounded degree, bounded tree-width, and nowhere density. The sparsification conjecture predicts that the (possibly dense) monadically stable graph classes are exactly those that can be logically encoded by first-order (FO) transductions in the (always sparse) nowhere dense classes. So far this conjecture has been verified for several special cases, such as for classes of bounded shrub-depth, and for the monadically stable fragments of bounded (linear) clique-width, twin-width, and merge-width. In this work we propose the existential positive sparsification conjecture, predicting that the more restricted co-matching-free, monadically stable classes are exactly those that can be transduced from nowhere dense classes using only existential positive FO formulas. While the general conjecture remains open, we verify its truth for all known special cases of the original conjecture. Even stronger, we find the sparse preimages as subgraphs of the dense input graphs. As a key ingredient, we introduce a new combinatorial operation, called subflip, that arises as the natural co-matching-free analog of the flip operation, which is a central tool in the characterization of monadic stability. Using subflips, we characterize the co-matching-free fragment of monadic stability by appropriate strengthenings of the known flip-flatness and flipper game characterizations for monadic stability. In an attempt to generalize our results to the more expressive MSO logic, we discover (rediscover?) that on relational structures (existential) positive MSO has the same expressive power as (existential) positive FO.

Algorithmic meta-theorems explain the tractability of large classes of computational problems by linking logical expressibility with structural graph properties. While extensions of first-order logic such as FO+dp admit efficient model checking on graph classes excluding a fixed topological minor, comparable results for richer fragments of CMSO were previously unknown. We further develop the framework of Sau, Stamoulis, and Thilikos [SODA 2025] for fragmenting CMSO via annotated graph parameters, which restrict set quantification to vertex sets satisfying bounded structural conditions. Following this approach, we identify a fragment of CMSO, namely the one defined by allowing quantification only over sets having what we call low monodimensionality, that generalizes several previously-known logics and we show that model checking for this fragment, enhanced with the disjoint-paths predicate, is fixed-parameter tractable on topological-minor-free graph classes. Such classes essentially delimit the tractability for this logic on subgraph-closed classes. As a consequence, our results lift several known algorithmic meta-theorems beyond first-order logic to the topological-minor-free setting.

We study solution discovery, where the goal is to obtain a feasible solution to a problem from an initial configuration by a bounded sequence of local moves. In many applications, however, the graph that defines which vertex sets are feasible is not the same as the graph that governs how tokens, agents, or resources may move. Existing models such as token sliding and token jumping typically do not distinguish the problem graph and the movement graph. Motivated by this mismatch, we introduce a directed weighted two-graph model that cleanly separates feasibility from movement. A problem graph specifies the desired combinatorial objects, while a movement graph specifies admissible relocations and their costs. This yields a flexible framework that captures asymmetry, heterogeneous movement constraints, and weighted transitions, while subsuming classical discovery models as special cases. We investigate this model through Path Discovery and Shortest Path Discovery, where the task is to realize a vertex set containing an s-t-path or a shortest s-t-path in the problem graph. These problems are particularly natural in applications, since directed and weighted shortest paths are among the most fundamental algorithmic primitives. At the same time, previous work has already shown that discovery can be computationally hard even when the underlying optimization problem is easy. Our results show that this phenomenon persists, and becomes especially rich, in the two-graph setting. We obtain a detailed complexity picture, identifying tractable cases as well as strong hardness results.

The dynamics of real-world applications and systems require efficient methods for improving infeasible solutions or restoring corrupted ones by making modifications to the current state of a system in a restricted way. We propose a new framework of solution discovery via reconfiguration for constructing a feasible solution for a given problem by executing a sequence of small modifications starting from a given state or configuration. Our framework integrates and formalizes different aspects of classical local search, reoptimization, and combinatorial reconfiguration. We exemplify our framework on a multitude of fundamental combinatorial problems, namely Vertex Cover , Independent Set , Dominating Set , and Coloring . We study the classical as well as the parameterized complexity of the solution discovery variants of those problems and explore the boundary between tractable and intractable instances.

The 10th iteration of the of the Parameterized Algorithms and Computational Experiments challenge (PACE) 2025 was devoted to engineer algorithms solving the Dominating Set problem as well as the Hitting Set problem. In contrast to the last iterations, these problems are (under standard assumptions) not fixed-parameter tractable (fpt) in general. However, restricting the structure of the input (e.g. to planar graphs or degenerate graphs for Dominating Set, or to set systems with sets of bounded size for Hitting Set) renders these problems fpt. Following the spirit of the last iterations of the PACE challenge, there is an exact track and a heuristic track for each problem; each track coming with a benchmark set of 100 public instances and 100 private instances. Overall, the PACE 2025 had 71 participants from 25 teams, 13 countries, and 3 continents. In this report, we briefly describe the setup of the challenge, the selection of benchmark instances, as well as the ranking of the participating teams. We also briefly outline the approaches used in the submitted solvers.

In the Dominated Cluster Deletion problem, we are given an undirected graph G and integers k and d and the question is to decide whether there exists a set of at most k vertices whose removal results in a graph in which each connected component has a dominating set of size at most d. In the Elimination Distance to Dominated Clusters problem, we are again given an undirected graph G and integers k and d and the question is to decide whether we can recursively delete vertices up to depth k such that each remaining connected component has a dominating set of size at most d. Bentert et al. [Bentert et al., MFCS 2024] recently provided an almost complete classification of the parameterized complexity of Dominated Cluster Deletion with respect to the parameters k, d, c, and Δ, where c and Δ are the degeneracy, and the maximum degree of the input graph, respectively. In particular, they provided a non-uniform algorithm with running time f(k,d)⋅ n^{𝒪(d)}. They left as an open problem whether the problem is fixed-parameter tractable with respect to the parameter k + d + c. We provide a uniform algorithm running in time f(k,d)⋅ n^{𝒪(d)} for both Dominated Cluster Deletion and Elimination Distance to Dominated Clusters. We furthermore show that both problems are FPT when parameterized by k+d+𝓁, where 𝓁 is the semi-ladder index of the input graph, a parameter that is upper bounded and may be much smaller than the degeneracy c, positively answering the open question of Bentert et al. We further complete the picture by providing an almost full classification for the parameterized complexity and kernelization complexity of Elimination Distance to Dominated Clusters. The one difficult base case that remains open is whether Treedepth (the case d = 0) is NP-hard on graphs of bounded maximum degree.

Abstract We study reduction rules for Directed Feedback Vertex Set (DFVS) on directed graphs without long cycles. A DFVS instance without cycles longer than d naturally corresponds to an instance of d -Hitting Set, however, enumerating all cycles in an n-vertex graph and then kernelizing the resulting d -Hitting Set instance can be too costly, as already enumerating all cycles can take time $$\Omega (n^d)$$ Ω ( n d ) . To the best of our knowledge, the kernelization of DFVS on graphs without long cycles has not been studied in the literature, except for very restricted cases, e.g., for tournaments, in which all induced cycles are of length three. We show that the natural reduction rule to delete all vertices and edges that do not lie on induced cycles cannot be implemented efficiently, that is, it is W[1]-hard (with respect to parameter d) to decide if a vertex or edge lies on an induced cycle of length at most d even on graphs that become acyclic after the deletion of a single vertex or edge. Based on different reduction rules we then show how to compute a kernel with at most $$2^dk^d$$ 2 d k d vertices and at most $$d^{3d}k^d$$ d 3 d k d induced cycles of length at most d (which however, cannot be enumerated efficiently), where k is the size of a minimum directed feedback vertex set. We then study classes of graphs whose underlying undirected graphs have bounded expansion or are nowhere dense. These are very general classes of sparse graphs, containing e.g. classes excluding a minor or a topological minor. We prove that for every class $$\mathscr {C} $$ C with bounded expansion there is a function $$f_\mathscr {C} (d)$$ f C ( d ) such that for graphs $$G\in \mathscr {C} $$ G ∈ C without induced cycles of length greater than d we can compute a kernel with $$f_\mathscr {C} (d)\cdot k$$ f C ( d ) · k vertices in time $$f_\mathscr {C} (d)\cdot n^{\mathcal {O}(1)}$$ f C ( d ) · n O ( 1 ) . For every nowhere dense class $$\mathscr {C} $$ C there is a function $$f_\mathscr {C} (d,\varepsilon )$$ f C ( d , ε ) such that for graphs $$G\in \mathscr {C} $$ G ∈ C without induced cycles of length greater than d we can compute a kernel with $$f_\mathscr {C} (d,\varepsilon )\cdot k^{1+\varepsilon }$$ f C ( d , ε ) · k

We study various aspects of the first-order transduction quasi-order on graph classes, which provides a way of measuring the relative complexity of graph classes based on whether one can encode the other using a formula of first-order (FO) logic. In contrast with the conjectured simplicity of the transduction quasi-order for monadic second-order logic, the FO-transduction quasi-order is very complex, and many standard properties from structural graph theory and model theory naturally appear in it. We prove a local normal form for transductions among other general results and constructions, which we illustrate via several examples and via the characterizations of the transductions of some simple classes. We then turn to various aspects of the quasi-order, including the (non-)existence of minimum and maximum classes for certain properties, the strictness of the pathwidth hierarchy, the fact that the quasi-order is not a lattice, and the role of weakly sparse classes in the quasi-order.

Bounding the number of reversals in a counter machine is one of the most prominent restrictions to achieve decidability of the reachability problem. Given this success, we explore whether this notion can be relaxed while retaining decidability. To this end, we introduce the notion of an f-reversal-bounded counter machine for a monotone function f: ℕ → ℕ. In such a machine, every run of length n makes at most f(n) reversals. Our first main result is a dichotomy theorem: We show that for every monotone function f, one of the following holds: Either (i) f grows so slowly that every f-reversal bounded counter machine is already k-reversal bounded for some constant k or (ii) f belongs to Ω(log(n)) and reachability in f-reversal bounded counter machines is undecidable. This shows that classical reversal bounding already captures the decidable cases of f-reversal bounding for any monotone function f. The key technical ingredient is an analysis of the growth of small solutions of iterated compositions of Presburger-definable constraints. In our second contribution, we investigate whether imposing f-reversal boundedness improves the complexity of the reachability problem in vector addition systems with states (VASS). Here, we obtain an analogous dichotomy: We show that either (i) f grows so slowly that every f-reversal-bounded VASS is already k-reversal-bounded for some constant k or (ii) f belongs to Ω(n) and the reachability problem for f-reversal-bounded VASS remains Ackermann-complete. This result is proven using run amalgamation in VASS. Overall, our results imply that classical restriction of reversal boundedness is a robust one.

In the recently introduced framework of solution discovery via reconfiguration [Fellows et al., ECAI 2023], we are given an initial configuration of $k$ tokens on a graph and the question is whether we can transform this configuration into a feasible solution (for some problem) via a bounded number $b$ of small modification steps. In this work, we study solution discovery variants of polynomial-time solvable problems, namely Spanning Tree Discovery, Shortest Path Discovery, Matching Discovery, and Vertex/Edge Cut Discovery in the unrestricted token addition/removal model, the token jumping model, and the token sliding model. In the unrestricted token addition/removal model, we show that all four discovery variants remain in P. For the toking jumping model we also prove containment in P, except for Vertex/Edge Cut Discovery, for which we prove NP-completeness. Finally, in the token sliding model, almost all considered problems become NP-complete, the exception being Spanning Tree Discovery, which remains polynomial-time solvable. We then study the parameterized complexity of the NP-complete problems and provide a full classification of tractability with respect to the parameters solution size (number of tokens) $k$ and transformation budget (number of steps) $b$. Along the way, we observe strong connections between the solution discovery variants of our base problems and their (weighted) rainbow variants as well as their red-blue variants with cardinality constraints.

In the solution discovery variant of a vertex (edge) subset problem Π on graphs, we are given an initial configuration of tokens on the vertices (edges) of an input graph G together with a budget b. The question is whether we can transform this configuration into a feasible solution of Π on G with at most b modification steps. We consider the token sliding variant of the solution discovery framework, where each modification step consists of sliding a token to an adjacent vertex (edge). The framework of solution discovery was recently introduced by Fellows et al. [ECAI 2023] and for many solution discovery problems the classical as well as the parameterized complexity has been established. In this work, we study the kernelization complexity of the solution discovery variants of Vertex Cover, Independent Set, Dominating Set, Shortest Path, Matching, and Vertex Cut with respect to the parameters number of tokens k, discovery budget b, as well as structural parameters such as pathwidth.

Disjoint-paths logic, denoted FO+ DP, extends first-order logic (FO) with atomic predicates dpk[(x1, y1), …, (xk, yk)], expressing the existence of internally vertex-disjoint paths between xi and yi, for 1 ≤ i ≤ k. We prove that for every graph class excluding some fixed graph as a topological minor, the model checking problem for FO+ DP is fixed-parameter tractable. This extends the model checking algorithm of Golovach et al. [SODA 2023] for FO+ DP for minor-closed graph classes. It also essentially settles the question of tractable model checking for this logic on subgraph-closed classes, since the problem is hard on subgraph-closed classes not excluding a topological minor (assuming a further mild condition on efficiency of encoding).

Abstract In the Token Sliding problem we are given a graph G and two independent sets $$I_s$$ I s and $$I_t$$ I t in G of size $$k \ge 1$$ k ≥ 1 . The goal is to decide whether there exists a sequence $$\langle I_1, I_2, \ldots , I_\ell \rangle $$ ⟨ I 1 , I 2 , … , I ℓ ⟩ of independent sets such that for all $$j \in \{1,\ldots , \ell - 1\}$$ j ∈ { 1 , … , ℓ - 1 } the set $$I_j$$ I j is an independent set of size k, $$I_1 = I_s$$ I 1 = I s , $$I_\ell = I_t$$ I ℓ = I t and $$I_j \triangle I_{j + 1} = \{u, v\} \in E(G)$$ I j ▵ I j + 1 = { u , v } ∈ E ( G ) . Intuitively, we view each independent set as a collection of tokens placed on the vertices of the graph. Then, the problem asks whether there exists a sequence of independent sets that transforms $$I_s$$ I s into $$I_t$$ I t where at each step we are allowed to slide one token from a vertex to a neighboring vertex. In this paper, we focus on the parameterized complexity of Token Sliding parameterized by k. As shown by Bartier et al. (Algorithmica 83(9):2914–2951, 2021. https://doi.org/10.1007/s00453-021-00848-1 ), the problem is -hard on graphs of girth four or less, and the authors posed the question of whether there exists a constant $$p \ge 5$$ p ≥ 5 such that the problem becomes fixed-parameter tractable on graphs of girth at least p. We answer their question positively and prove that the problem is indeed fixed-parameter tractable on graphs of girth five or more, which establishes a full classification of the tractability of Token Sliding parameterized by the number of tokens based on the girth of the input graph.

We study the graph parameter elimination distance to bounded degree, which was introduced by Bulian and Dawar in their study of the parameterized complexity of the graph isomorphism problem. We prove that the problem is fixed-parameter tractable on planar graphs, that is, there exists an algorithm that given a planar graph G and integers d, k decides in time f( k, d) · n c for a computable function f and constant c whether the elimination distance of G to the class of degree d graphs is at most k.

Inspired by a width invariant on permutations defined by Guillemot and Marx, Bonnet, Kim, Thomass\'e, and Watrigant introduced the twin-width of graphs, which is a parameter describing its structural complexity. This invariant has been further extended to binary structures, in several (basically equivalent) ways. We prove that a class of binary relational structures (that is: edge-colored partially directed graphs) has bounded twin-width if and only if it is a first-order transduction of a~proper permutation class. As a by-product, we show that every class with bounded twin-width contains at most $2^{O(n)}$ pairwise non-isomorphic $n$-vertex graphs.

The dynamics of real-world applications and systems require efficient methods for improving infeasible solutions or restoring corrupted ones by making modifications to the current state of a system in a restricted way. We propose a new framework of solution discovery via reconfiguration for constructing a feasible solution for a given problem by executing a sequence of small modifications starting from a given state. Our framework integrates different aspects of classical local search, reoptimization, and combinatorial reconfiguration. We exemplify our framework on a multitude of fundamental combinatorial problems, namely VERTEX COVER, INDEPENDENT SET, DOMINATING SET, and COLORING. We study the classical as well as the parameterized complexity of the solution discovery variants of those problems and explore the boundary between tractable and intractable instances.

Monadically stable and monadically NIP classes of structures were initially studied in the context of model theory and defined in logical terms. They have recently attracted attention in the area of structural graph theory, as they generalize notions such as nowhere denseness, bounded cliquewidth, and bounded twinwidth. Our main result is the - to the best of our knowledge first - purely combinatorial characterization of monadically stable classes of graphs, in terms of a property dubbed flip-flatness. A class $\mathcal{C}$ of graphs is flip-flat if for every fixed radius $r$, every sufficiently large set of vertices of a graph $G \in \mathcal{C}$ contains a large subset of vertices with mutual distance larger than $r$, where the distance is measured in some graph $G'$ that can be obtained from $G$ by performing a bounded number of flips that swap edges and non-edges within a subset of vertices. Flip-flatness generalizes the notion of uniform quasi-wideness, which characterizes nowhere dense classes and had a key impact on the combinatorial and algorithmic treatment of nowhere dense classes. To obtain this result, we develop tools that also apply to the more general monadically NIP classes, based on the notion of indiscernible sequences from model theory. We show that in monadically stable and monadically NIP classes indiscernible sequences impose a strong combinatorial structure on their definable neighborhoods. All our proofs are constructive and yield efficient algorithms.

A class of graphs $\mathscr{C}$ is monadically stable if for any unary expansion $\widehat{\mathscr{C}}$ of $\mathscr{C}$, one cannot interpret, in first-order logic, arbitrarily long linear orders in graphs from $\widehat{\mathscr{C}}$. It is known that nowhere dense graph classes are monadically stable; these encompass most of the studied concepts of sparsity in graphs, including graph classes that exclude a fixed topological minor. On the other hand, monadic stability is a property expressed in purely model-theoretic terms and hence it is also suited for capturing structure in dense graphs. For several years, it has been suspected that one can create a structure theory for monadically stable graph classes that mirrors the theory of nowhere dense graph classes in the dense setting. In this work we provide a step in this direction by giving a characterization of monadic stability through the Flipper game: a game on a graph played by Flipper, who in each round can complement the edge relation between any pair of vertex subsets, and Connector, who in each round localizes the game to a ball of bounded radius. This is an analog of the Splitter game, which characterizes nowhere dense classes of graphs (Grohe, Kreutzer, and Siebertz, J.ACM'17). We give two different proofs of our main result. The first proof uses tools from model theory, and it exposes an additional property of monadically stable graph classes that is close in spirit to definability of types. Also, as a byproduct, we give an alternative proof of the recent result of Braunfeld and Laskowski (arXiv 2209.05120) that monadic stability for graph classes coincides with existential monadic stability. The second proof relies on the recently introduced notion of flip-wideness (Dreier, Mählmann, Siebertz, and Toruńczyk, ICALP 2023) and provides an efficient algorithm to compute Flipper's moves in a winning strategy.

A class of graphs is structurally nowhere dense if it can be constructed from a nowhere dense class by a first-order transduction. Structurally nowhere dense classes vastly generalize nowhere dense classes and constitute important examples of monadically stable classes. We show that the first-order model checking problem is fixed-parameter tractable on every structurally nowhere dense class of graphs.

First-order logic (FO) can express many algorithmic problems on graphs, such as the independent set and dominating set problem parameterized by solution size. However, FO cannot express the very simple algorithmic question whether two vertices are connected. We enrich FO with connectivity predicates that are tailored to express algorithmic graph problems that are commonly studied in parameterized algorithmics. By adding the atomic predicates connk(x,y,z_1,..., zk) that hold true in a graph if there exists a path between (the valuations of)xandyafter (the valuations of)z1,..., zkhave been deleted, we obtainseparator logicFO + conn. We show that separator logic can express many interesting problems, such as the feedback vertex set problem and elimination distance problems to first-order definable classes. Denote by FO + connkthe fragment of separator logic that is restricted to connectivity predicates with at mostk + 2variables (that is, at mostkdeletions), we show that FO + connk + 1is strictly more expressive than FO + connkfor allk ≥ 0. We then study the limitations of separator logic and prove that it cannot express planarity, and, in particular, not the disjoint paths problem. We obtain the strongerdisjoint-paths logicFO + DP by adding the atomic predicates disjoint-pathsk[(x1, y1),..., (xk, yk) that evaluate to true if there are internally vertex-disjoint paths between (the valuations of)xiandyifor all 1 ≤ i ≤ k. Disjoint-paths logic can express the disjoint paths problem, the problem of (topological) minor containment, the problem of hitting (topological) minors, and many more. Again, we show that the fragments FO + DPkthat use predicates for at mostkdisjoint paths form a strict hierarchy of expressiveness. Finally, we compare the expressive power of the new logics with that of transitive-closure logics and monadic second-order logic.

Logical transductions provide a very useful tool to encode classes of structures inside other classes of structures. In this paper we study first-order (FO) transductions and the quasiorder they induce on infinite classes of finite graphs. Surprisingly, this quasiorder is very complex, though shaped by the locality properties of first-order logic. This contrasts with the conjectured simplicity of the monadic second order (MSO) transduction quasiorder. We first establish a local normal form for FO transductions, which is of independent interest. Then we prove that the quotient partial order is a bounded distributive join-semilattice, and that the subposet of additive classes is also a bounded distributive join-semilattice. The FO transduction quasiorder has a great expressive power, and many well studied class properties can be defined using it. We apply these structural properties to prove, among other results, that FO transductions of the class of paths are exactly perturbations of classes with bounded bandwidth, that the local variants of monadic stability and monadic dependence are equivalent to their (standard) non-local versions, and that the classes with pathwidth at most k, for k ≥ 1 form a strict hierarchy in the FO transduction quasiorder.

First-order logic (FO) can express many algorithmic problems on graphs, such as the independent set and dominating set problem parameterized by solution size. On the other hand, FO cannot express the very simple algorithmic question whether two vertices are connected. We enrich FO with connectivity predicates that are tailored to express algorithmic graph properties that are commonly studied in parameterized algorithmics. By adding the atomic predicates conn_k(x,y,z_1,…,z_k) that hold true in a graph if there exists a path between (the valuations of) x and y after (the valuations of) z_1,…,z_k have been deleted, we obtain separator logic FO+conn. We show that separator logic can express many interesting problems such as the feedback vertex set problem and elimination distance problems to first-order definable classes. Denote by FO+conn_k the fragment of separator logic that is restricted to connectivity predicates with at most k+2 variables (that is, at most k deletions). We show that FO+conn_{k+1} is strictly more expressive than FO+conn_k for all k ≥ 0. We then study the limitations of separator logic and prove that it cannot express planarity, and, in particular, not the disjoint paths problem. We obtain the stronger disjoint-paths logic FO+DP by adding the atomic predicates disjoint-paths_k[(x_1,y_1),…,(x_k,y_k)] that evaluate to true if there are internally vertex-disjoint paths between (the valuations of) x_i and y_i for all 1 ≤ i ≤ k. Disjoint-paths logic can express the disjoint paths problem, the problem of (topological) minor containment, the problem of hitting (topological) minors, and many more. Again we show that the fragments FO+DP_k that use predicates for at most k disjoint paths form a strict hierarchy of expressiveness. Finally, we compare the expressive power of the new logics with that of transitive-closure logics and monadic second-order logic.

We introduce a new data structure for answering connectivity queries in undirected graphs subject to batched vertex failures. Precisely, given any graph G and integer parameter k, we can in fixed-parameter time construct a data structure that can later be used to answer queries of the form: "are vertices s and t connected via a path that avoids vertices u₁,…, u_k?" in time 2^𝒪(k). In the terminology of the literature on data structures, this gives the first deterministic data structure for connectivity under vertex failures where for every fixed number of failures, all operations can be performed in constant time. With the aim to understand the power and the limitations of our new techniques, we prove an algorithmic meta theorem for the recently introduced separator logic, which extends first-order logic with atoms for connectivity under vertex failures. We prove that the model-checking problem for separator logic is fixed-parameter tractable on every class of graphs that exclude a fixed topological minor. We also show a weak converse. This implies that from the point of view of parameterized complexity, under standard complexity theoretical assumptions, the frontier of tractability of separator logic is almost exactly delimited by classes excluding a fixed topological minor. The backbone of our proof relies on a decomposition theorem of Cygan, Lokshtanov, Pilipczuk, Pilipczuk, and Saurabh [SICOMP '19], which provides a tree decomposition of a given graph into bags that are unbreakable. Crucially, unbreakability allows to reduce separator logic to plain first-order logic within each bag individually. Guided by this observation, we design our model-checking algorithm using dynamic programming over the tree decomposition, where the transition at each bag amounts to running a suitable model-checking subprocedure for plain first-order logic. This approach is robust enough to provide also an extension to efficient enumeration of answers to a query expressed in separator logic.

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We present an exact solver for the DFVS, submitted for the exact track of the Parameterized Algorithms and Computational Experiments challenge (PACE) in 2022. The solver heavily relies on data reduction (known from the literature and new reduction rules). The instances are then further processed by integer linear programming approaches. We implemented the algorithm in the scope of a student project at the University of Bremen.

A strong backdoor in a formula $ϕ$ of propositional logic to a tractable class $\mathcal{C}$ of formulas is a set $B$ of variables of $ϕ$ such that every assignment of the variables in $B$ results in a formula from $\mathcal{C}$. Strong backdoors of small size or with a good structure, e.g. with small backdoor treewidth, lead to efficient solutions for the propositional satisfiability problem SAT. In this paper we propose the new notion of recursive backdoors, which is inspired by the observation that in order to solve SAT we can independently recurse into the components that are created by partial assignments of variables. The quality of a recursive backdoor is measured by its recursive backdoor depth. Similar to the concept of backdoor treewidth, recursive backdoors of bounded depth include backdoors of unbounded size that have a certain treelike structure. However, the two concepts are incomparable and our results yield new tractability results for SAT.

We study two notions of being well-structured for classes of graphs that are inspired by classic model theory. A class of graphs is monadically stable if it is impossible to define arbitrarily long linear orders in vertex-colored graphs from using a fixed first-order formula. Similarly, monadic dependence corresponds to the impossibility of defining all graphs in this way. Examples of monadically stable graph classes are nowhere dense classes, which provide a robust theory of sparsity. Examples of monadically dependent classes are classes of bounded rankwidth (or equivalently, bounded cliquewidth), which can be seen as a dense analog of classes of bounded treewidth. Thus, monadic stability and monadic dependence extend classical structural notions for graphs by viewing them in a wider, model-theoretical context. We explore this emerging theory by proving the following: 1) A class of graphs is a first-order transduction of a class with bounded treewidth if and only if has bounded rankwidth and a stable edge relation (i.e. graphs from exclude some half-graph as a semi-induced subgraph). 2) If a class of graphs is monadically dependent and not monadically stable, then has in fact an unstable edge relation. As a consequence, we show that classes with bounded rankwidth excluding some half-graph as a semi-induced subgraph are linearly χ-bounded. Our proofs are effective and lead to polynomial time algorithms.

In a reconfiguration version of an optimization problem $\mathcal{Q}$ the input is an instance of $\mathcal{Q}$ and two feasible solutions $S$ and $T$. The objective is to determine whether there exists a step-by-step transformation between $S$ and $T$ such that all intermediate steps also constitute feasible solutions. In this work, we study the parameterized complexity of the \textsc{Connected Dominating Set Reconfiguration} problem (\textsc{CDS-R)}. It was shown in previous work that the \textsc{Dominating Set Reconfiguration} problem (\textsc{DS-R}) parameterized by $k$, the maximum allowed size of a dominating set in a reconfiguration sequence, is fixed-parameter tractable on all graphs that exclude a biclique $K_{d,d}$ as a subgraph, for some constant $d \geq 1$. We show that the additional connectivity constraint makes the problem much harder, namely, that \textsc{CDS-R} is \textsf{W}$[1]$-hard parameterized by $k+\ell$, the maximum allowed size of a dominating set plus the length of the reconfiguration sequence, already on $5$-degenerate graphs. On the positive side, we show that \textsc{CDS-R} parameterized by $k$ is fixed-parameter tractable, and in fact admits a polynomial kernel on planar graphs.

We study the graph parameter elimination distance to bounded degree, which was introduced by Bulian and Dawar in their study of the parameterized complexity of the graph isomorphism problem. We prove that the problem is fixed-parameter tractable on planar graphs, that is, there exists an algorithm that given a planar graph G and integers d and k decides in time f(k,d)⋅ n^c for a computable function f and constant c whether the elimination distance of G to the class of degree d graphs is at most k.

Classes with bounded rankwidth are MSO-transductions of trees and classes with bounded linear rankwidth are MSO-transductions of paths. These results show a strong link between the properties of these graph classes considered from the point of view of structural graph theory and from the point of view of finite model theory. We take both views on classes with bounded linear rankwidth and prove structural and model theoretic properties of these classes: 1) Graphs with linear rankwidth at most r are linearly χ-bounded. Actually, they have bounded c-chromatic number, meaning that they can be colored with f (r) colors, each color inducing a cograph. 2) Based on a Ramsey-like argument, we prove for every proper hereditary family of graphs (like cographs) that there is a class with bounded rankwidth that does not have the property that graphs in it can be colored by a bounded number of colors, each inducing a subgraph in . 3) For a class with bounded linear rankwidth the following conditions are equivalent: a) is stable, b) excludes some half-graph as a semi-induced subgraph, c) is a first-order transduction of a class with bounded pathwidth. These results open the perspective to study classes admitting low linear rankwidth covers.

We study the model-checking problem for first- and monadic second-order logic on finite relational structures. The problem of verifying whether a formula of these logics is true on a given structure is considered intractable in general, but it does become tractable on interesting classes of structures, such as on classes whose Gaifman graphs have bounded treewidth. In this article, we continue this line of research and study model-checking for first- and monadic second-order logic in the presence of an ordering on the input structure. We do so in two settings: the general ordered case, where the input structures are equipped with a fixed order or successor relation, and the order-invariant case, where the formulas may resort to an ordering, but their truth must be independent of the particular choice of order. In the first setting we show very strong intractability results for most interesting classes of structures. In contrast, in the order-invariant case we obtain tractability results for order-invariant monadic second-order formulas on the same classes of graphs as in the unordered case. For first-order logic, we obtain tractability of successor-invariant formulas on classes whose Gaifman graphs have bounded expansion. Furthermore, we show that model-checking for order-invariant first-order formulas is tractable on coloured posets of bounded width.

The notion of bounded expansion captures uniform sparsity of graph classes and renders various algorithmic problems that are hard in general tractable. In particular, the model-checking problem for first-order logic is fixed-parameter tractable over such graph classes. With the aim of generalizing such results to dense graphs, we introduce classes of graphs with structurally bounded expansion , defined as first-order transductions of classes of bounded expansion. As a first step towards their algorithmic treatment, we provide their characterization analogous to the characterization of classes of bounded expansion via low treedepth covers (or colorings), replacing treedepth by its dense analogue called shrubdepth.

For p ∊ ℕ, a coloring λ of the vertices of a graph G is p-centered if for every connected subgraph H of G, either H receives more than p colors under λ or there is a color that appears exactly once in H. Centered colorings play an important role in the theory of sparse graphs introduced by Nešetřil and Ossona de Mendez [27], as they structurally characterize classes of bounded expansion, one of the key notions in this theory. More precisely, a class of graphs ℒ has bounded expansion if and only if there is a function f : ℕ → ℕ such that every graph G ∊ ℒ for every p ∊ ℕ admits a p-centered coloring with at most f(p) colors. Unfortunately, known proofs of the existence of such colorings yield large upper bounds on the function f governing the number of colors needed, even for as simple classes as planar graphs. We prove that every Kt-minor-free graph admits a p-centered coloring with O(pg(t)) colors for some function g. In the special case that the graph is embeddable in a fixed surface Σ we show that it admits a p-centered coloring with O(p19) colors, with the degree of the polynomial independent of the genus of Σ. This provides the first polynomial upper bounds on the number of colors needed in p-centered colorings of graphs drawn from proper minor-closed classes, which answers an open problem posed by Dvořák [1]. As an algorithmic application, we use our main result to prove that if ℒ is a fixed proper minor-closed class of graphs, then given graphs H and G, on p and n vertices, respectively, where G ∊ ℒ, it can be decided whether H is a subgraph of G in time 2O(p log p). nO(1) and space nO(1).

We consider a generic algorithmic paradigm that we call progressive exploration, which can be used to develop simple and efficient parameterized graph algorithms. We identify two model-theoretic properties that lead to efficient progressive algorithms, namely variants of the Helly property and stability. We demonstrate our approach by giving linear-time fixed-parameter algorithms for the Distance-r Dominating Set problem (parameterized by the solution size) in a wide variety of restricted graph classes, such as powers of nowhere dense classes, map graphs, and (for r=1) biclique-free graphs. Similarly, for the Distance-r Independent Set problem the technique can be used to give a linear-time fixed-parameter algorithm on any nowhere dense class. Despite the simplicity of the method, in several cases our results extend known boundaries of tractability for the considered problems and improve the best known running times.

The notions of bounded expansion [Nešetřil and Ossona de Mendez, 2008] and nowhere denseness [Nešetřil and Ossona de Mendez, 2011], introduced by Nešetřil and Ossona de Mendez as structural measures for undirected graphs, have been applied very successfully in algorithmic graph theory. We study the corresponding notions of directed bounded expansion and nowhere crownfulness on directed graphs, introduced by Kreutzer and Tazari [Kreutzer and Tazari, 2012]. The classes of directed graphs having those properties are very general classes of sparse directed graphs, as they include, on one hand, all classes of directed graphs whose underlying undirected class has bounded expansion, such as planar, bounded-genus, and H-minor-free graphs, and on the other hand, they also contain classes whose underlying undirected class is not even nowhere dense. We show that many of the algorithmic tools that were developed for undirected bounded expansion classes can, with some care, also be applied in their directed counterparts, and thereby we highlight a rich algorithmic structure theory of directed bounded expansion and nowhere crownful classes.

For $\alpha > 1$, an $\alpha$-approximate (bi)kernel is a polynomial-time algorithm that takes as input an instance $(I, k)$ of a problem $\mathcal{Q}$ and outputs an instance $(I',k')$ (of a problem $\mathcal{Q}'$) of size bounded by a function of $k$ such that, for every $c\geq 1$, a $c$-approximate solution for the new instance can be turned into a $(c\cdot\alpha)$-approximate solution of the original instance in polynomial time. This framework of lossy kernelization was recently introduced by Lokshtanov and co-authors. We study Connected Dominating Set (and its distance-$r$ variant) parameterized by solution size on sparse graph classes like biclique-free graphs, classes of bounded expansion, and nowhere dense classes. We prove that for every $\alpha>1$, Connected Dominating Set admits a polynomial-size $\alpha$-approximate (bi)kernel on all the aforementioned classes. Our results are in sharp contrast to the kernelization complexity of Connected Dominating Set, which is known to not admit a polynomial kernel even on $2$-degenerate graphs and graphs of bounded expansion, unless ${NP} \subseteq \textsf{coNP/poly}$. We complement our results by the following conditional lower bound. We show that if a class $\mathcal{C}$ is somewhere dense and closed under taking subgraphs, then for some value of $r\in \mathbb{N}$ there cannot exist an $\alpha$-approximate bi-kernel for the (Connected) Distance-$r$ Dominating Set problem on $\mathcal{C}$ for any $\alpha>1$ (assuming ${FPT}\neq{W}[1]$).

The Minimum Dominating Set (MDS) problem is a fundamental and challenging problem in distributed computing. While it is well known that minimum dominating sets cannot be well approximated locally on general graphs, in recent years there has been much progress on computing good local approximations on sparse graphs and in particular on planar graphs. In this article, we study distributed and deterministic MDS approximation algorithms for graph classes beyond planar graphs. In particular, we show that existing approximation bounds for planar graphs can be lifted to bounded genus graphs and more general graphs, which we call locally embeddable graphs, and present (1) a local constant-time, constant-factor MDS approximation algorithm on locally embeddable graphs, and (2) a local O (log * n )-time (1+ϵ)-approximation scheme for any ϵ > 0 on graphs of bounded genus. Our main technical contribution is a new analysis of a slightly modified variant of an existing algorithm by Lenzen et al. [21]. Interestingly, unlike existing proofs for planar graphs, our analysis does not rely on direct topological arguments but on combinatorial density arguments only.

Nowhere dense classes of graphs [21, 22] are very general classes of uniformly sparse graphs with several seemingly unrelated characterisations. From an algorithmic perspective, a characterisation of these classes in terms of uniform quasi-wideness , a concept originating in finite model theory, has proved to be particularly useful. Uniform quasi-wideness is used in many fpt-algorithms on nowhere dense classes. However, the existing constructions showing the equivalence of nowhere denseness and uniform quasi-wideness imply a non-elementary blow up in the parameter dependence of the fpt-algorithms, making them infeasible in practice. As a first main result of this article, we use tools from logic, in particular from a sub-field of model theory known as stability theory, to establish polynomial bounds for the equivalence of nowhere denseness and uniform quasi-wideness. A powerful method in parameterized complexity theory is to compute a problem kernel in a pre-computation step, that is, to reduce the input instance in polynomial time to a sub-instance of size bounded in the parameter only (independently of the input graph size). Our new tools allow us to obtain for every fixed radius r ∈ N a polynomial kernel for the distance- r dominating set problem on nowhere dense classes of graphs. This result is particularly interesting, as it implies that for every class C of graphs that is closed under taking subgraphs, the distance- r dominating set problem admits a kernel on C for every value of r if, and only if, it already admits a polynomial kernel for every value of r (under the standard assumption of parameterized complexity theory that FPT ≠ W[2]).

We prove that for every class ℒ of graphs with effectively bounded expansion, given a first-order sentence φ and an n-element structure A whose Gaifman graph belongs to ℒ, the question whether φ holds in A can be decided by a family of AC-circuits of size f(φ) · nc and depth f(φ) + c log n, where f is a computable function and c is a universal constant. This places the model-checking problem for classes of bounded expansion in the parameterized circuit complexity class para-AC1. On the route to our result we prove that the basic decomposition toolbox for classes of bounded expansion, including orderings with bounded weak coloring numbers and low treedepth decompositions, can be computed in para-AC1.

We prove that for every class of graphs ℒ which is nowhere dense, as defined by Nešetřil and Ossona de Mendez [28, 29], and for every first order formula φ(x, y), whenever one draws a graph G ∈ ℒ and a subset of its nodes A, the number of subsets of A|y| which are of the form {u ∈ A|y|: G |= φ(ū, v)} for some valuation ū of x in G is bounded by O(|A||x|ε), for every ε > 0. This provides optimal bounds on the VC-density of first-order definable set systems in nowhere dense graph classes. We also give two new proofs of upper bounds on quantities in nowhere dense classes which are relevant for their logical treatment. Firstly, we provide a new proof of the fact that nowhere dense classes are uniformly quasi-wide, implying explicit, polynomial upper bounds on the functions relating the two notions. Secondly, we give a new combinatorial proof of the result of Adler and Adler [1] stating that every nowhere dense class of graphs is stable. In contrast to the previous proofs of the above results, our proofs are completely finitistic and constructive, and yield explicit and computable upper bounds on quantities related to uniform quasi-wideness (margins) and stability (ladder indices).

\noindent We provide a new constant factor approximation algorithm for the (connected) \mboxdistance- r dominating set problem on graph classes of bounded expansion. Classes of bounded expansion include many familiar classes of sparse graphs such as planar graphs and graphs with excluded (topological) minors, and notably, these classes form the most general subgraph closed classes of graphs for which a sequential constant factor approximation algorithm for the distance- r dominating set problem is currently known. Our algorithm can be implemented in the \congestbc model of distributed computing and uses $Øof(r^2 łog n)$ communication rounds. % Our techniques, which may be of independent interest, are based on a distributed computation of sparse neighborhood covers of small radius on bounded expansion classes. We show how to compute an r -neighborhood cover of radius~$2r$ and overlap $f(r)$ on every class of bounded expansion in $Øof(r^2łog n)$ communication rounds for some function~ f .% in the \congestbc model. % Finally, we show how to use the greater power of the łocal model to turn any distance- r dominating set into a constantly larger connected distance- r dominating set in $3r+1$ rounds on any class of bounded expansion. Combining this algorithm, e.g., with the constant factor approximation algorithm for dominating sets on planar graphs of Lenzen et al.\ gives a constant factor approximation algorithm for connected dominating sets on planar graphs in a constant number of rounds in the łocal model, where the approximation ratio is only $6$ times larger than that of Lenzen et al.'s algorithm.

In [M. Grohe, S. Kreutzer, and S. Siebertz, J. ACM, 64 (2017), 17] it was shown that nowhere dense classes of graphs admit sparse neighborhood covers of small degree. We show that a monotone graph class admits sparse neighborhood covers if and only if it is nowhere dense. The existence of such covers for nowhere dense classes is established through bounds on so-called weak coloring numbers. The core results of this paper are various lower and upper bounds on the weak coloring numbers and other, closely related, generalized coloring numbers. We prove tight bounds for these numbers on graphs of bounded treewidth. We clarify and tighten the relation between the density of shallow minors and the various generalized coloring numbers. These upper bounds are complemented by new, stronger exponential lower bounds on the weak and strong coloring numbers, and by superpolynomial lower bounds on the weak coloring numbers on classes of polynomial expansion. Finally, we show that computing weak $r$-coloring numbers is NP-complete for all $r\geq 3$.

We prove that whenever G is a graph from a nowhere dense graph class C, and A is a subset of vertices of G, then the number of subsets of A that are realized as intersections of A with r-neighborhoods of vertices of G is at most f(r,eps)|A|^(1+eps), where r is any positive integer, eps is any positive real, and f is a function that depends only on the class C. This yields a characterization of nowhere dense classes of graphs in terms of neighborhood complexity, which answers a question posed by [Reidl et al., CoRR, 2016]. As an algorithmic application of the above result, we show that for every fixed integer r, the parameterized Distance-r Dominating Set problem admits an almost linear kernel on any nowhere dense graph class. This proves a conjecture posed by [Drange et al., STACS 2016], and shows that the limit of parameterized tractability of Distance-r Dominating Set on subgraph-closed graph classes lies exactly on the boundary between nowhere denseness and somewhere denseness.

A successor-invariant first-order formula is a formula that has access to an auxiliary successor relation on a structure's universe, but the model relation is independent of the particular interpretation of this relation. It is well known that successor-invariant formulas are more expressive on finite structures than plain first-order formulas without a successor relation. This naturally raises the question whether this increase in expressive power comes at an extra cost to solve the model-checking problem, that is, the problem to decide whether a given structure together with some (and hence every) successor relation is a model of a given formula. It was shown earlier that adding successor-invariance to first-order logic essentially comes at no extra cost for the model-checking problem on classes of finite structures whose underlying Gaifman graph is planar [1], excludes a fixed minor [2] or a fixed topological minor [3], [4]. In this work we show that the model-checking problem for successor-invariant formulas is fixed-parameter tractable on any class of finite structures whose underlying Gaifman graphs form a class of bounded expansion. Our result generalises all earlier results and comes close to the best tractability results on nowhere dense classes of graphs currently known for plain first-order logic.

Nowhere dense classes of graphs [21, 22] are very general classes of uniformly sparse graphs with several seemingly unrelated characterisations. From an algorithmic perspective, a characterisation of these classes in terms of uniform quasi-wideness, a concept originating in finite model theory, has proved to be particularly useful. Uniform quasi-wideness is used in many fpt-algorithms on nowhere dense classes. However, the existing constructions showing the equivalence of nowhere denseness and uniform quasi-wideness imply a non-elementary blow up in the parameter dependence of the fpt-algorithms, making them infeasible in practice. As a first main result of this paper, we use tools from logic, in particular from a sub-field of model theory known as stability theory, to establish polynomial bounds for the equivalence of nowhere denseness and uniform quasi-wideness. As an algorithmic application of our new methods, we obtain for every fixed value of r ∊ ℕ a polynomial kernel for the distance-r dominating set problem on nowhere dense classes of graphs. This is particularly interesting, as it implies that for every subgraph-closed class C, the distance-r dominating set problem admits a kernel on C for every value of r if, and only if, it admits a polynomial kernel for every value of r (under the standard assumption of parameterized complexity theory that FPT ≠ W[2]). Finally, we demonstrate how to use the new methods to improve the parameter dependence of many fixed- parameter algorithms. As an example we provide a single exponential parameterized algorithm for the Connected Dominating Set problem on nowhere dense graph classes.

Bounded expansion and nowhere dense graph classes, introduced by Nesetril and Ossona de Mendez, form a large variety of classes of uniformly sparse graphs which includes the class of planar graphs, actually all classes with excluded minors, and also bounded degree graphs. Since their initial definition it was shown that these graph classes can be defined in many equivalent ways: by generalised colouring numbers, neighbourhood complexity, sparse neighbourhood covers, a game known as the splitter game, and many more. We study the corresponding concepts for directed graphs. We show that the densities of bounded depth directed minors and bounded depth topological minors relate in a similar way as in the undirected case. We provide a characterisation of bounded expansion classes by a directed version of the generalised colouring numbers. As an application we show how to construct sparse directed neighbourhood covers and how to approximate directed distance-r dominating sets on classes of bounded expansion. On the other hand, we show that linear neighbourhood complexity does not characterise directed classes of bounded expansion.

Nowhere dense graph classes, introduced by Nešetřil and Ossona de Mendez [2010, 2011], form a large variety of classes of “sparse graphs” including the class of planar graphs, actually all classes with excluded minors, and also bounded degree graphs and graph classes of bounded expansion. We show that deciding properties of graphs definable in first-order logic is fixed-parameter tractable on nowhere dense graph classes (parameterized by the length of the input formula). At least for graph classes closed under taking subgraphs, this result is optimal: it was known before that for all classes C of graphs closed under taking subgraphs, if deciding first-order properties of graphs in C is fixed-parameter tractable, then C must be nowhere dense (under a reasonable complexity theoretic assumption). As a by-product, we give an algorithmic construction of sparse neighborhood covers for nowhere dense graphs. This extends and improves previous constructions of neighborhood covers for graph classes with excluded minors. At the same time, our construction is considerably simpler than those. Our proofs are based on a new game-theoretic characterization of nowhere dense graphs that allows for a recursive version of locality-based algorithms on these classes. On the logical side, we prove a “rank-preserving” version of Gaifman’s locality theorem.

The generalised colouring numbers $\mathrm{adm}_r(G)$, $\mathrm{col}_r(G)$, and $\mathrm{wcol}_r(G)$ were introduced by Kierstead and Yang as generalisations of the usual colouring number, also known as the degeneracy of a graph, and have since then found important applications in the theory of bounded expansion and nowhere dense classes of graphs, introduced by Nešetřil and Ossona de Mendez. In this paper, we study the relation of the colouring numbers with two other measures that characterise nowhere dense classes of graphs, namely with uniform quasi-wideness, studied first by Dawar et al. in the context of preservation theorems for first-order logic, and with the splitter game, introduced by Grohe et al. We show that every graph excluding a fixed topological minor admits a universal order, that is, one order witnessing that the colouring numbers are small for every value of $r$. Finally, we use our construction of such orders to give a new proof of a result of Eickmeyer and Kawarabayashi, showing that the model-checking problem for successor-invariant first-order formulas is fixed-parameter tractable on classes of graphs with excluded topological minors.

The Minimum Dominating Set (MDS) problem is not only one of the most fundamental problems in distributed computing, it is also one of the most challenging ones. While it is well-known that minimum dominating sets cannot be approximated locally on general graphs, over the last years, several breakthroughs have been made on computing local approximations on sparse graphs.

In cops and robber games a number of cops tries to capture a robber in a graph. A variant of these games on undirected graphs characterises tree width by the least number of cops needed to win. We consider cops and robber games on digraphs and width measures (such as DAG-width, directed tree width or D-width) corresponding to them. All of them generalise tree width and the game characterising it. For the DAG-width game we prove that the problem to decide the minimal number of cops required to capture the robber (which is the same as deciding DAG-width), is PSPACE-complete, in contrast to most other similar games. We also show that the cop-monotonicity cost for directed tree width games cannot be bounded by any function. As a consequence, D-width is not bounded in directed tree width, refuting a conjecture by Safari. A large number of directed width measures generalising tree width has been proposed in the literature. However, only very little was known about the relation between them, in particular about whether classes of digraphs of bounded width in one measure have bounded width in another. In this paper we establish an almost complete order among the most prominent width measures with respect to mutual boundedness.

Nowhere dense graph classes, introduced by Nešetřil and Ossona de Mendez [30], form a large variety of classes of "sparse graphs" including the class of planar graphs, actually all classes with excluded minors, and also bounded degree graphs and graph classes of bounded expansion.

Nowhere dense classes of graphs were introduced by Nesetril and Ossona de Mendez as a model for "sparsity" in graphs. It turns out that nowhere dense classes of graphs can be characterised in many different ways and have been shown to be equivalent to other concepts studied in areas such as (finite) model theory. Therefore, the concept of nowhere density seems to capture a natural property of graph classes generalising for example classes of graphs which exclude a fixed minor, have bounded degree or bounded local tree-width. In this paper we give a self-contained introduction to the concept of nowhere dense classes of graphs focussing on the various ways in which they can be characterised. We also briefly sketch algorithmic applications these characterisations have found in the literature.

We investigate the logical resources required to maintain knowledge about a property of a finite structure that undergoes an ongoing series of local changes such as insertion or deletion of tuples to basic relations. Our framework is closely related to the Dyn-FO-framework of Patnaik and Immerman and the FOIES-framework of Dong, Libkin, Su and Wong, and also builds on work of Weber and Schwentick. We assume that the dynamic process starts with an arbitrary, nonempty structure, but in contrast to previous work, we assume that, in general, structures are unordered. We show how to modify known dynamic algorithms for symmetric reachability, bipartiteness, k-edge connectivity and more, to work also without an order and with dynamic processes starting at an arbitrary graph. A history independent dynamic system (also called deterministic or memoryless) is one that maintains all auxiliary information independent of the update order. In 1997, Dong and Su posed the problem whether there exist history independent dynamic systems with FO-updates for symmetric reachability or bipartiteness. We give a positive answer to this question. We further show that there is a history independent system for tree isomorphism with FO+C-updates. On the other hand we show that on unordered structures first-order logic is too weak to maintain enough information to answer the equal cardinality query and the tree isomorphism query dynamically.

Model-checking for first- and monadic second-order logic in the context of graphs has received considerable attention in the literature. It is well-known that the problem of verifying whether a formula of these logics is true in a graph is computationally intractable but it does become tractable on interesting classes of graphs such as classes of bounded tree-width. In this paper we continue this line of research but study model checking for first- and monadic second-order logic in the presence of an ordering on the input structure. We do so in two settings: the general ordered case, where the input structures are equipped with a fixed order or successor relation, and the order invariant case, where the formulas may resort to an ordering but their truth must be independent of the particular choice of order. In the first setting we show very strong intractability results for most interesting classes of graphs. In contrast, in order invariant case we obtain tractability results for order invariant monadic second-order logic on the same classes of graphs as in the unordered case. For first-order logic, we obtain tractability of successor-invariant FO on planar graphs.