arXiv subjects

Nikolas Mählmann

Publications and source records attributed to Nikolas Mählmann.

17 recordsLinked to original sources

arXiv2026

Hereditary 2-WQO Graph Classes Have Bounded Clique-Width

A graph class is $k$-WQO if its $k$-labeled graphs are well-quasi-ordered under label-preserving induced subgraph embeddings. We show that every hereditary graph class that is $2$-WQO has bounded clique-width. Combined with the recent result of Dumas and Lopez, this confirms a long-standing conjecture of Pouzet: A hereditary graph class is $2$-WQO if and only if it is $k$-WQO for all $k\geq 2$, if and only if it is $\forall$-WQO, that is, its labeled graphs are well-quasi-ordered for every possible well-quasi-ordered label set. Our proof builds on a recent structure/non-structure dichotomy for the model theoretic notion of monadic dependence by Dreier, Mählmann, and Toruńczyk. Through the non-structure characterization by forbidden induced subgraphs, we show that every hereditary $2$-WQO graph class is monadically dependent. Leveraging the Ramsey-theoretic structural properties provided by monadic dependence, we then establish bounded clique-width by ruling out the existence of large well-linked sets, which are the canonical obstructions for clique-width.

math.CO↗

arXiv2026

Fatness and Flatness

Fat minors are the metric analog of graph minors that are tailored to the analysis of metric (edge-weighted) graphs and, more generally, metric spaces having a suitable notion of shortest paths. Despite a large interest in this notion, not much is known about the structure of metric graphs excluding a fixed fat minor. We prove that if a metric graph $G$ excludes a fixed graph $H$ as a $δ$-fat minor, for some $δ>0$, then $G$ enjoys the metric analog of flatness (aka uniform quasi-wideness) - a structural property from the field of Sparsity. In essence, our flatness result says that for any $α\geq β$ large enough compared to $δ$, in every large enough set $A$ in $G$ one can find a sizable subset $B$ that becomes $α$-scattered after removing a bounded number of balls of radius $β$. We call this property drill-flatness. Notably, the proof only relies on excluding shallow fat minors: every branch set has radius at most $2α$. As a corollary, we prove that metric graphs that exclude a fixed $δ$-fat minor have bounded $\varepsilon$-scatter dimension if we consider only $\varepsilon$-scatters at distances large enough compared to $δ$. By combining this with the results of Abbasi et al. [FOCS 2023], we infer that the $k$-Center problem on instances excluding $H$ as a $δ$-fat minor admits an approximation algorithm that finds a solution of cost at most $(1+\varepsilon)\cdot\mathsf{OPT}+{\cal O}(δ/\varepsilon^2)$ in time ${\cal O}_{H,\varepsilon}(n^{{\cal O}(1)})$. This is one of the first algorithmic results for general fat-minor-free metrics. We also study drill-flatness in hereditary classes of (unweighted) graphs, where we obtain a characterization equating drill-flatness with excluding shallow induced minors. This is an induced analog of the equivalence between flatness and nowhere denseness - one of central results of Sparsity.

math.CO↗

arXiv2026

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes

Monadic dependence is a proposed structural dividing line for fixed-parameter tractability of first-order model checking on hereditary graph classes. A graph class is \emph{monadically dependent} if the class of all graphs cannot be interpreted in its vertex-colored members using a fixed first-order formula. We prove two structural consequences of monadic dependence. First, every monadically dependent class has \emph{almost linear neighborhood complexity}: for every graph $G$ in the class and every set $A\subseteq V(G)$, the family $\{N_G(v)\cap A : v\in V(G)\}$ has size $|A|^{1+o(1)}$. Second, every $n$-vertex graph in a monadically dependent class has radius-1 merge-width $n^{o(1)}$. Here, merge-width is the decomposition parameter of Dreier and Toruńczyk based on construction sequences; its radius-$r$ version measures local reachability among parts through already resolved pairs. This settles the radius-1 case of the conjectured connection between monadic dependence and almost bounded merge-width and provides the first decomposition-based structural description of monadically dependent graph classes. Our proof is algorithmic: we give an $\mathcal{O}(n^5)$-time algorithm that, given an $n$-vertex graph $G$ such that $|\{N_G(v)\cap A : v\in V(G)\}|\le O(|A|^d)$ for every $A\subseteq V(G)$, computes a construction sequence witnessing radius-1 merge-width $\mathcal{O}(n^{1-1/d}\log n)$.

cs.DM↗

arXiv2026

A Note on Constructive Canonical Splitter Strategies in Nowhere Dense Graph Classes

The radius-$r$ splitter game is played on a graph $G$ between two players: Splitter and Connector. In each round, Connector selects a vertex $v$, and the current game arena is restricted to the radius-$r$ neighborhood of $v$. Then Splitter removes a vertex from this restricted subgraph. The game ends, and Splitter wins, when the arena becomes empty. Splitter aims to end the game as quickly as possible, while Connector tries to prolong it for as long as possible. The splitter game was introduced by Grohe, Kreutzer and Siebertz to characterize nowhere dense graph classes. They showed that a class $\mathscr{C}$ of graphs is nowhere dense if and only if for every radius $r$ there exists a number $k$ such that Splitter has a strategy on every $G\in \mathscr{C}$ to win the radius-$r$ splitter game in at most $k$ rounds. It was recently proved by Ohlmann et al. that for every nowhere dense class $\mathscr{C}$ and every radius $r$ there are only a bounded number of possible Splitter moves that are progressing, that is, moves that lead to an arena where Splitter can win in one less round. The proof of Ohlmann et al. is based on the compactness theorem and does not give a constructive bound on the number of progressing moves. In this work, we give a simple constructive proof, showing that if Splitter can force a win in the radius-$r$ game in $k$ rounds, then there are at most $(2r+1)^{\,2^{k-1}-1}$ progressing moves.

cs.LO↗

arXiv2026

Flips and Merge-Width in Sparse Graphs

A flip of a graph is obtained by complementing the edge relation within a set of vertices. Flips are typically used to separate vertices in a graph, by increasing the distances between them. We show that in $K_{t,t}$-free graphs, every short sequence of flips can be simulated by a short sequence of vertex deletions that achieves a similar degree of separation: distances in the resulting graph are, up to a factor of three, at least as large as those obtained after the flips. This result provides a simple and uniform explanation of an emerging pattern in structural graph theory and finite model theory: the $K_{t,t}$-free fragment of a tameness notion for dense graphs often coincides with a tameness notion for sparse graphs. As immediate applications, we recover the following known equivalences. In the $K_{t,t}$-free setting, the dense notions (1) bounded shrub-depth, (2) bounded clique-width, (3) bounded flip-width, (4) monadic dependence, respectively, coincide with the sparse notions (1) bounded tree-depth, (2) bounded tree-width, (3) bounded expansion, and (4) no-where dense-ness. Furthermore, we reprove the result by Dreier and Toruńczyk (STOC 2025) stating that $K_{t,t}$-free classes of bounded merge-width have bounded expansion. Our proof provides explicit bounds and is direct, as it shows how to construct strong coloring orders (witnesses of bounded expansion) from merge sequences (witnesses of bounded merge-width). Along the way, we identify a new family of graph parameters, dubbed separation-width, that is sandwiched between the strong and weak coloring numbers, and is closely related to the merge-width parameters. We provide evidence that this family of graph parameters, apparently overlooked in the literature, may play a fundamental role in the study of sparse graphs.

math.CO↗

arXiv2026

The Parameterized Complexity of Independent Set and More when Excluding a Half-Graph, Co-Matching, or Matching

A theorem of Ding, Oporowski, Oxley, and Vertigan implies that any sufficiently large twin-free graph contains a large matching, a co-matching, or a half-graph as a semi-induced subgraph. The sizes of these unavoidable patterns are measured by the matching index, co-matching index, and half-graph index of a graph. Consequently, graph classes can be organized into the eight classes determined by which of the three indices are bounded. We completely classify the parameterized complexity of Independent Set, Clique, and Dominating Set across all eight of these classes. For this purpose, we first derive multiple tractability and hardness results from the existing literature, and then proceed to fill the identified gaps. Among our novel results, we show that Independent Set is fixed-parameter tractable on every graph class where the half-graph and co-matching indices are simultaneously bounded. Conversely, we construct a graph class with bounded half-graph index (but unbounded co-matching index), for which the problem is W[1]-hard. For the W[1]-hard cases of our classification, we review the state of approximation algorithms. Here, we contribute an approximation algorithm for Independent Set on classes of bounded half-graph index.

cs.CC↗

arXiv2026

Existential Positive Transductions of Sparse Graphs

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.

cs.DM↗

arXiv2025

Separability Properties of Monadically Dependent Graph Classes

A graph class $\mathcal C$ is monadically dependent if one cannot interpret all graphs in colored graphs from $\mathcal C$ using a fixed first-order interpretation. We prove that monadically dependent classes can be exactly characterized by the following property, which we call flip-separability: for every $r\in \mathbb{N}$, $\varepsilon>0$, and every graph $G\in \mathcal{C}$ equipped with a weight function on vertices, one can apply a bounded (in terms of $\mathcal{C},r,\varepsilon$) number of flips (complementations of the adjacency relation on a subset of vertices) to $G$ so that in the resulting graph, every radius-$r$ ball contains at most an $\varepsilon$-fraction of the total weight. On the way to this result, we introduce a robust toolbox for working with various notions of local separations in monadically dependent classes.

math.CO↗

arXiv2025

Flipper games for monadically stable graph classes

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.

cs.LO↗

arXiv2025

Forbidden Induced Subgraphs for Bounded Shrub-Depth and the Expressive Power of MSO

The graph parameter shrub-depth is a dense analog of tree-depth. We characterize classes of bounded shrub-depth by forbidden induced subgraphs. The obstructions are well-controlled flips of large half-graphs and of disjoint unions of many long paths. Applying this characterization, we show that on every hereditary class of unbounded shrub-depth, MSO is more expressive than FO. This confirms a conjecture of [Gajarský and Hliněný; LMCS 2015] who proved that on classes of bounded shrub-depth FO and MSO have the same expressive power. Combined, the two results fully characterize the hereditary classes on which FO and MSO coincide, answering an open question by [Elberfeld, Grohe, and Tantau; LICS 2012]. Our work is inspired by the notion of stability from model theory. A graph class C is MSO-stable, if no MSO-formula can define arbitrarily long linear orders in graphs from C. We show that a hereditary graph class is MSO-stable if and only if it has bounded shrub-depth. As a key ingredient, we prove that every hereditary class of unbounded shrub-depth FO-interprets the class of all paths. This improves upon a result of [Ossona de Mendez, Pilipczuk, and Siebertz; Eur. J. Comb. 2025] who showed the same statement for FO-transductions instead of FO-interpretations.

cs.LO↗

arXiv2024

Flip-Breakability: A Combinatorial Dichotomy for Monadically Dependent Graph Classes

A conjecture in algorithmic model theory predicts that the model-checking problem for first-order logic is fixed-parameter tractable on a hereditary graph class if and only if the class is monadically dependent. Originating in model theory, this notion is defined in terms of logic, and encompasses nowhere dense classes, monadically stable classes, and classes of bounded twin-width. Working towards this conjecture, we provide the first two combinatorial characterizations of monadically dependent graph classes. This yields the following dichotomy. On the structure side, we characterize monadic dependence by a Ramsey-theoretic property called flip-breakability. This notion generalizes the notions of uniform quasi-wideness, flip-flatness, and bounded grid rank, which characterize nowhere denseness, monadic stability, and bounded twin-width, respectively, and played a key role in their respective model checking algorithms. Natural restrictions of flip-breakability additionally characterize bounded treewidth and cliquewidth and bounded treedepth and shrubdepth. On the non-structure side, we characterize monadic dependence by explicitly listing few families of forbidden induced subgraphs. This result is analogous to the characterization of nowhere denseness via forbidden subdivided cliques, and allows us to resolve one half of the motivating conjecture: First-order model checking is AW[$*$]-hard on every hereditary graph class that is monadically independent. The result moreover implies that hereditary graph classes which are small, have almost bounded twin-width, or have almost bounded flip-width, are monadically dependent. Lastly, we lift our result to also obtain a combinatorial dichotomy in the more general setting of monadically dependent classes of binary structures.

math.CO↗

arXiv2023

First-Order Model Checking on Monadically Stable Graph Classes

A graph class $\mathscr{C}$ is called monadically stable if one cannot interpret, in first-order logic, arbitrary large linear orders in colored graphs from $\mathscr{C}$. We prove that the model checking problem for first-order logic is fixed-parameter tractable on every monadically stable graph class. This extends the results of [Grohe, Kreutzer, and Siebertz; J. ACM '17] for nowhere dense classes and of [Dreier, Mählmann, and Siebertz; STOC '23] for structurally nowhere dense classes to all monadically stable classes. As a complementary hardness result, we prove that for every hereditary graph class $\mathscr{C}$ that is edge-stable (excludes some half-graph as a semi-induced subgraph) but not monadically stable, first-order model checking is $\mathrm{AW}[*]$-hard on $\mathscr{C}$, and $\mathrm{W}[1]$-hard when restricted to existential sentences. This confirms, in the special case of edge-stable classes, an on-going conjecture that the notion of monadic NIP delimits the tractability of first-order model checking on hereditary classes of graphs. For our tractability result, we first prove that monadically stable graph classes have almost linear neighborhood complexity. Using this, we construct sparse neighborhood covers for monadically stable classes, which provides the missing ingredient for the algorithm of [Dreier, Mählmann, and Siebertz; STOC '23]. The key component of this construction is the usage of orders with low crossing number [Welzl; SoCG '88], a tool from the area of range queries. For our hardness result, we prove a new characterization of monadically stable graph classes in terms of forbidden induced subgraphs. We then use this characterization to show that in hereditary classes that are edge-stable but not monadically stable, one can effectively interpret the class of all graphs using only existential formulas.

cs.LO↗

arXiv2023

Indiscernibles and Flatness in Monadically Stable and Monadically NIP Classes

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.

cs.LO↗

arXiv2023

First-Order Model Checking on Structurally Sparse Graph Classes

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. Our result builds on a recently developed game-theoretic characterization of monadically stable graph classes. As a second key ingredient of independent interest, we provide a polynomial-time algorithm for approximating weak neighborhood covers (on general graphs). We combine the two tools into a recursive locality-based model checking algorithm. This algorithm is efficient on every monadically stable graph class admitting flip-closed sparse weak neighborhood covers, where flip-closure is a mild additional assumption. Thereby, establishing efficient first-order model checking on monadically stable classes is reduced to proving the existence of flip-closed sparse weak neighborhood covers on these classes - a purely combinatorial problem. We complete the picture by proving the existence of the desired covers for structurally nowhere dense classes: we show that every structurally nowhere dense class can be sparsified by contracting local sets of vertices, enabling us to lift the existence of covers from sparse classes.

cs.LO↗

arXiv2022

Combinatorial and Algorithmic Aspects of Monadic Stability

Nowhere dense classes of graphs are classes of sparse graphs with rich structural and algorithmic properties, however, they fail to capture even simple classes of dense graphs. Monadically stable classes, originating from model theory, generalize nowhere dense classes and close them under transductions, i.e. transformations defined by colorings and simple first-order interpretations. In this work we aim to extend some combinatorial and algorithmic properties of nowhere dense classes to monadically stable classes of finite graphs. We prove the following results. - In monadically stable classes the Ramsey numbers $R(s,t)$ are bounded from above by $\mathcal{O}(t^{s-1-δ})$ for some $δ>0$, improving the bound $R(s,t)\in \mathcal{O}(t^{s-1}/(\log t)^{s-1})$ known for general graphs and the bounds known for $k$-stable graphs when $s\leq k$. - For every monadically stable class $\mathcal{C}$ and every integer $r$, there exists $δ> 0$ such that every graph $G \in \mathcal{C}$ that contains an $r$-subdivision of the biclique $K_{t,t}$ as a subgraph also contains $K_{t^δ,t^δ}$ as a subgraph. This generalizes earlier results for nowhere dense graph classes. - We obtain a stronger regularity lemma for monadically stable classes of graphs. - Finally, we show that we can compute polynomial kernels for the independent set and dominating set problems in powers of nowhere dense classes. Formerly, only fixed-parameter tractable algorithms were known for these problems on powers of nowhere dense classes.

cs.DM↗

arXiv2022

Model Checking on Interpretations of Classes of Bounded Local Cliquewidth

We present a fixed-parameter tractable algorithm for first-order model checking on interpretations of graph classes with bounded local cliquewidth. Notably, this includes interpretations of planar graphs, and more generally, of classes of bounded genus. To obtain this result we develop a new tool which works in a very general setting of dependent classes and which we believe can be an important ingredient in achieving similar results in the future.

cs.DS↗

arXiv2021

Recursive Backdoors for SAT

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.

cs.DS↗