23 Search Results for "Gajarský, Jakub"

Document
DOI: 10.4230/LIPIcs.SoCG.2026.51
First-Order Logic and Twin-Width for Some Geometric Graphs

Authors: Colin Geniet, Gunwoo Kim, and Lucas Meijer

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
For some geometric graph classes, tractability of testing first-order formulas is precisely characterised by the graph parameter twin-width. This was first proved for interval graphs among others in [BCKKLT, IPEC '22], where the equivalence is called delineation, and more generally holds for circle graphs, rooted directed path graphs, and H-graphs when H is a forest. Delineation is based on the key idea that geometric graphs often admit natural vertex orderings, allowing to use the very rich theory of twin-width for ordered graphs. Answering two questions raised in their work, we prove that delineation holds for intersection graphs of non-degenerate axis-parallel unit segment graphs, but fails for visibility graphs of 1.5D terrains. We also prove delineation for intersection graphs of circular arcs.
Cite as

Colin Geniet, Gunwoo Kim, and Lucas Meijer. First-Order Logic and Twin-Width for Some Geometric Graphs. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 51:1-51:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{geniet_et_al:LIPIcs.SoCG.2026.51,
  author =	{Geniet, Colin and Kim, Gunwoo and Meijer, Lucas},
  title =	{{First-Order Logic and Twin-Width for Some Geometric Graphs}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{51:1--51:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.51},
  URN =		{urn:nbn:de:0030-drops-258575},
  doi =		{10.4230/LIPIcs.SoCG.2026.51},
  annote =	{Keywords: Twin-width, axis-parallel unit segment graphs, circular arc graphs, terrain visibility graphs, first-order logic, model checking, FPT}
}
Document
DOI: 10.4230/LIPIcs.CSL.2026.41
Weakly-Sparse and Strongly Flip-Flat Classes of Graphs Are Uniformly Almost-Wide

Authors: Fatemeh Ghasemi, Julien Grange, Mamadou Moustapha Kanté, and Florent Madelaine

Published in: LIPIcs, Volume 363, 34th EACSL Annual Conference on Computer Science Logic (CSL 2026)

Abstract
In this work we take a step towards characterising strongly flip-flat classes of graphs. Strong flip-flatness appears to be the analogue of uniform almost-wideness in the setting of dense classes of graphs. We prove that strongly flip-flat classes of graphs that are weakly sparse are indeed uniformly almost-wide.
Cite as

Fatemeh Ghasemi, Julien Grange, Mamadou Moustapha Kanté, and Florent Madelaine. Weakly-Sparse and Strongly Flip-Flat Classes of Graphs Are Uniformly Almost-Wide. In 34th EACSL Annual Conference on Computer Science Logic (CSL 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 363, pp. 41:1-41:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{ghasemi_et_al:LIPIcs.CSL.2026.41,
  author =	{Ghasemi, Fatemeh and Grange, Julien and Kant\'{e}, Mamadou Moustapha and Madelaine, Florent},
  title =	{{Weakly-Sparse and Strongly Flip-Flat Classes of Graphs Are Uniformly Almost-Wide}},
  booktitle =	{34th EACSL Annual Conference on Computer Science Logic (CSL 2026)},
  pages =	{41:1--41:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-411-6},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{363},
  editor =	{Guerrini, Stefano and K\"{o}nig, Barbara},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2026.41},
  URN =		{urn:nbn:de:0030-drops-254668},
  doi =		{10.4230/LIPIcs.CSL.2026.41},
  annote =	{Keywords: Almost-wide, Flip-flatness}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2025.30
A Graph Width Perspective on Partially Ordered Hamiltonian Paths and Cycles II: Vertex and Edge Deletion Numbers

Authors: Jesse Beisegel, Katharina Klost, Kristin Knorr, Fabienne Ratajczak, and Robert Scheffler

Published in: LIPIcs, Volume 358, 20th International Symposium on Parameterized and Exact Computation (IPEC 2025)

Abstract
We consider the problem of finding a Hamiltonian path or cycle with precedence constraints in the form of a partial order on the vertex set. We study the complexity for graph width parameters for which the ordinary problems Hamiltonian Path and Hamiltonian Cycle are in FPT. In particular, we focus on parameters that describe how many vertices and edges have to be deleted to become a member of a certain graph class. We show that the problems are W[1]-hard for such restricted cases as vertex distance to path and vertex distance to clique. We complement these results by showing that the problems can be solved in XP time for vertex distance to outerplanar and vertex distance to block. Furthermore, we present some FPT algorithms, e.g., for edge distance to block. Additionally, we prove para-NP-hardness when considered with the edge clique cover number.
Cite as

Jesse Beisegel, Katharina Klost, Kristin Knorr, Fabienne Ratajczak, and Robert Scheffler. A Graph Width Perspective on Partially Ordered Hamiltonian Paths and Cycles II: Vertex and Edge Deletion Numbers. In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 30:1-30:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{beisegel_et_al:LIPIcs.IPEC.2025.30,
  author =	{Beisegel, Jesse and Klost, Katharina and Knorr, Kristin and Ratajczak, Fabienne and Scheffler, Robert},
  title =	{{A Graph Width Perspective on Partially Ordered Hamiltonian Paths and Cycles II: Vertex and Edge Deletion Numbers}},
  booktitle =	{20th International Symposium on Parameterized and Exact Computation (IPEC 2025)},
  pages =	{30:1--30:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-407-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{358},
  editor =	{Agrawal, Akanksha and van Leeuwen, Erik Jan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2025.30},
  URN =		{urn:nbn:de:0030-drops-251623},
  doi =		{10.4230/LIPIcs.IPEC.2025.30},
  annote =	{Keywords: Hamiltonian path, Hamiltonian cycle, partial order, graph width parameter, parameterized complexity}
}
Document
DOI: 10.4230/LIPIcs.GD.2025.16
Structural Parameterizations of k-Planarity

Authors: Tatsuya Gima, Yasuaki Kobayashi, and Yuto Okada

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract
The concept of k-planarity is extensively studied in the context of Beyond Planarity. A graph is k-planar if it admits a drawing in the plane in which each edge is crossed at most k times. The local crossing number of a graph is the minimum integer k such that it is k-planar. The problem of determining whether an input graph is 1-planar is known to be NP-complete even for near-planar graphs [Cabello and Mohar, SIAM J. Comput. 2013], that is, the graphs obtained from planar graphs by adding a single edge. Moreover, the local crossing number is hard to approximate within a factor 2 - ε for any ε > 0 [Urschel and Wellens, IPL 2021]. To address this computational intractability, Bannister, Cabello, and Eppstein [JGAA 2018] investigated the parameterized complexity of the case of k = 1, particularly focusing on structural parameterizations on input graphs, such as treedepth, vertex cover number, and feedback edge number. In this paper, we extend their approach by considering the general case k ≥ 1 and give (tight) parameterized upper and lower bound results. In particular, we strengthen the aforementioned lower bound results to subclasses of constant-treewidth graphs: we show that testing 1-planarity is NP-complete even for near-planar graphs with feedback vertex set number at most 3 and pathwidth at most 4, and the local crossing number is hard to approximate within any constant factor for graphs with feedback vertex set number at most 2.
Cite as

Tatsuya Gima, Yasuaki Kobayashi, and Yuto Okada. Structural Parameterizations of k-Planarity. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 16:1-16:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{gima_et_al:LIPIcs.GD.2025.16,
  author =	{Gima, Tatsuya and Kobayashi, Yasuaki and Okada, Yuto},
  title =	{{Structural Parameterizations of k-Planarity}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{16:1--16:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.16},
  URN =		{urn:nbn:de:0030-drops-250021},
  doi =		{10.4230/LIPIcs.GD.2025.16},
  annote =	{Keywords: 1-planar graphs, local crossing number, beyond planarity, parameterized complexity, kernelization}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2025.13
Solving Partial Dominating Set and Related Problems Using Twin-Width

Authors: Jakub Balabán, Daniel Mock, and Peter Rossmanith

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
Partial vertex cover and partial dominating set are two well-investigated optimization problems. While they are W[1]-hard on general graphs, they have been shown to be fixed-parameter tractable on many sparse graph classes, including nowhere-dense classes. In this paper, we demonstrate that these problems are also fixed-parameter tractable with respect to the twin-width of a graph. Indeed, we establish a more general result: every graph property that can be expressed by a logical formula of the form ϕ≡∃ x₁⋯ ∃ x_k ∑_{α ∈ I} #y ψ_α(x₁,…,x_k,y) ≥ t, where ψ_α is a quantifier-free formula for each α ∈ I, t is an arbitrary number, and #y is a counting quantifier, can be evaluated in time f(d,k)n, where n is the number of vertices and d is the width of a contraction sequence that is part of the input. In addition to the aforementioned problems, this includes also connected partial dominating set and independent partial dominating set.
Cite as

Jakub Balabán, Daniel Mock, and Peter Rossmanith. Solving Partial Dominating Set and Related Problems Using Twin-Width. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 13:1-13:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{balaban_et_al:LIPIcs.MFCS.2025.13,
  author =	{Balab\'{a}n, Jakub and Mock, Daniel and Rossmanith, Peter},
  title =	{{Solving Partial Dominating Set and Related Problems Using Twin-Width}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{13:1--13:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.13},
  URN =		{urn:nbn:de:0030-drops-241203},
  doi =		{10.4230/LIPIcs.MFCS.2025.13},
  annote =	{Keywords: Partial Dominating Set, Partial Vertex Cover, meta-algorithm, counting logic, twin-width}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2025.65
Parameterized Spanning Tree Congestion

Authors: Michael Lampis, Valia Mitsou, Edouard Nemery, Yota Otachi, Manolis Vasilakis, and Daniel Vaz

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
In this paper we study the Spanning Tree Congestion problem, where we are given an undirected graph G = (V,E) and are asked to find a spanning tree T of minimum maximum congestion. Here, the congestion of an edge e ∈ T is the number of edges uv ∈ E such that the (unique) path from u to v in T traverses e. We consider this well-studied NP-hard problem from the point of view of (structural) parameterized complexity and obtain the following results: - We resolve a natural open problem by showing that Spanning Tree Congestion is not FPT parameterized by treewidth (under standard assumptions). More strongly, we present a generic reduction which applies to (almost) any parameter of the form "vertex-deletion distance to class 𝒞", thus obtaining W[1]-hardness for more restricted parameters, including tree-depth plus feedback vertex set, or incomparable to treewidth, such as twin cover. Via a slight tweak of the same reduction we also show that the problem is NP-complete on graphs of modular-width 4. - Even though it is known that Spanning Tree Congestion remains NP-hard on instances with only one vertex of unbounded degree, it is currently open whether the problem remains hard on bounded-degree graphs. We resolve this question by showing NP-hardness on graphs of maximum degree 8. - Complementing the problem’s W[1]-hardness for treewidth, we formulate an algorithm that runs in time roughly {(k+w)}^{𝒪(w)}, where k is the desired congestion and w the treewidth, improving a previous argument for parameter k+w that was based on Courcelle’s theorem. This explicit algorithm pays off in two ways: it allows us to obtain an FPT approximation scheme for parameter treewidth, that is, a (1+ε)-approximation running in time roughly {(w/ε)}^{𝒪(w)}; and it leads to an exact FPT algorithm for parameter clique-width+k via a Win/Win argument. - Finally, motivated by the problem’s hardness for most standard structural parameters, we present FPT algorithms for several more restricted cases, namely, for the parameters vertex-deletion distance to clique; vertex integrity; and feedback edge set, in the latter case also achieving a single-exponential running time dependence on the parameter.
Cite as

Michael Lampis, Valia Mitsou, Edouard Nemery, Yota Otachi, Manolis Vasilakis, and Daniel Vaz. Parameterized Spanning Tree Congestion. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 65:1-65:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{lampis_et_al:LIPIcs.MFCS.2025.65,
  author =	{Lampis, Michael and Mitsou, Valia and Nemery, Edouard and Otachi, Yota and Vasilakis, Manolis and Vaz, Daniel},
  title =	{{Parameterized Spanning Tree Congestion}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{65:1--65:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.65},
  URN =		{urn:nbn:de:0030-drops-241724},
  doi =		{10.4230/LIPIcs.MFCS.2025.65},
  annote =	{Keywords: Parameterized Complexity, Treewidth, Graph Width Parameters}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2025.147
Separability Properties of Monadically Dependent Graph Classes

Authors: Édouard Bonnet, Samuel Braunfeld, Ioannis Eleftheriadis, Colin Geniet, Nikolas Mählmann, Michał Pilipczuk, Wojciech Przybyszewski, and Szymon Toruńczyk

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
A graph class 𝒞 is monadically dependent if one cannot interpret all graphs in colored graphs from 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 ∈ ℕ, ε > 0, and every graph G ∈ 𝒞 equipped with a weight function on vertices, one can apply a bounded (in terms of 𝒞,r,ε) 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 ε-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.
Cite as

Édouard Bonnet, Samuel Braunfeld, Ioannis Eleftheriadis, Colin Geniet, Nikolas Mählmann, Michał Pilipczuk, Wojciech Przybyszewski, and Szymon Toruńczyk. Separability Properties of Monadically Dependent Graph Classes. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 147:1-147:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bonnet_et_al:LIPIcs.ICALP.2025.147,
  author =	{Bonnet, \'{E}douard and Braunfeld, Samuel and Eleftheriadis, Ioannis and Geniet, Colin and M\"{a}hlmann, Nikolas and Pilipczuk, Micha{\l} and Przybyszewski, Wojciech and Toru\'{n}czyk, Szymon},
  title =	{{Separability Properties of Monadically Dependent Graph Classes}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{147:1--147:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.147},
  URN =		{urn:nbn:de:0030-drops-235246},
  doi =		{10.4230/LIPIcs.ICALP.2025.147},
  annote =	{Keywords: Structural graph theory, Monadic dependence}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2025.167
Forbidden Induced Subgraphs for Bounded Shrub-Depth and the Expressive Power of MSO

Authors: Nikolas Mählmann

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
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.
Cite as

Nikolas Mählmann. Forbidden Induced Subgraphs for Bounded Shrub-Depth and the Expressive Power of MSO. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 167:1-167:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{mahlmann:LIPIcs.ICALP.2025.167,
  author =	{M\"{a}hlmann, Nikolas},
  title =	{{Forbidden Induced Subgraphs for Bounded Shrub-Depth and the Expressive Power of MSO}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{167:1--167:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.167},
  URN =		{urn:nbn:de:0030-drops-235444},
  doi =		{10.4230/LIPIcs.ICALP.2025.167},
  annote =	{Keywords: Shrub-Depth, Forbidden Induced Subgraphs, MSO, Stability Theory}
}
Document
DOI: 10.4230/LIPIcs.STACS.2025.6
Twin-Width One

Authors: Jungho Ahn, Hugo Jacob, Noleen Köhler, Christophe Paul, Amadeus Reinald, and Sebastian Wiederrecht

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

Abstract
We investigate the structure of graphs of twin-width at most 1, and obtain the following results: - Graphs of twin-width at most 1 are permutation graphs. In particular they have an intersection model and a linear structure. - There is always a 1-contraction sequence closely following a given permutation diagram. - Based on a recursive decomposition theorem, we obtain a simple algorithm running in linear time that produces a 1-contraction sequence of a graph, or guarantees that it has twin-width more than 1. - We characterise distance-hereditary graphs based on their twin-width and deduce a linear time algorithm to compute optimal sequences on this class of graphs.
Cite as

Jungho Ahn, Hugo Jacob, Noleen Köhler, Christophe Paul, Amadeus Reinald, and Sebastian Wiederrecht. Twin-Width One. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 6:1-6:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{ahn_et_al:LIPIcs.STACS.2025.6,
  author =	{Ahn, Jungho and Jacob, Hugo and K\"{o}hler, Noleen and Paul, Christophe and Reinald, Amadeus and Wiederrecht, Sebastian},
  title =	{{Twin-Width One}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{6:1--6:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.6},
  URN =		{urn:nbn:de:0030-drops-228319},
  doi =		{10.4230/LIPIcs.STACS.2025.6},
  annote =	{Keywords: Twin-width, Hereditary graph classes, Intersection model}
}
Document
DOI: 10.4230/LIPIcs.STACS.2025.64
Cluster Editing on Cographs and Related Classes

Authors: Manuel Lafond, Alitzel López Sánchez, and Weidong Luo

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

Abstract
In the Cluster Editing problem, sometimes known as (unweighted) Correlation Clustering, we must insert and delete a minimum number of edges to achieve a graph in which every connected component is a clique. Owing to its applications in computational biology, social network analysis, machine learning, and others, this problem has been widely studied for decades and is still undergoing active research. There exist several parameterized algorithms for general graphs, but little is known about the complexity of the problem on specific classes of graphs. Among the few important results in this direction, if only deletions are allowed, the problem can be solved in polynomial time on cographs, which are the P₄-free graphs. However, the complexity of the broader editing problem on cographs is still open. We show that even on a very restricted subclass of cographs, the problem is NP-hard, W[1]-hard when parameterized by the number p of desired clusters, and that time n^o(p/log p) is forbidden under the ETH. This shows that the editing variant is substantially harder than the deletion-only case, and that hardness holds for the many superclasses of cographs (including graphs of clique-width at most 2, perfect graphs, circle graphs, permutation graphs). On the other hand, we provide an almost tight upper bound of time n^O(p), which is a consequence of a more general n^O(cw⋅p) time algorithm, where cw is the clique-width. Given that forbidding P₄s maintains NP-hardness, we look at {P₄, C₄}-free graphs, also known as trivially perfect graphs, and provide a cubic-time algorithm for this class.
Cite as

Manuel Lafond, Alitzel López Sánchez, and Weidong Luo. Cluster Editing on Cographs and Related Classes. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 64:1-64:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{lafond_et_al:LIPIcs.STACS.2025.64,
  author =	{Lafond, Manuel and L\'{o}pez S\'{a}nchez, Alitzel and Luo, Weidong},
  title =	{{Cluster Editing on Cographs and Related Classes}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{64:1--64:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.64},
  URN =		{urn:nbn:de:0030-drops-228895},
  doi =		{10.4230/LIPIcs.STACS.2025.64},
  annote =	{Keywords: Cluster editing, cographs, parameterized algorithms, clique-width, trivially perfect graphs}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2024.61
ℋ-Clique-Width and a Hereditary Analogue of Product Structure

Authors: Petr Hliněný and Jan Jedelský

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
We introduce a novel generalization of the notion of clique-width which aims to bridge the gap between classical hereditary width measures and the recently introduced graph product structure theory. Bounding the new H-clique-width, in the special case of H being the class of paths, is equivalent to admitting a hereditary (i.e., induced) product structure of a path times a graph of bounded clique-width. Furthermore, every graph admitting the usual (non-induced) product structure of a path times a graph of bounded tree-width, has bounded H-clique-width and, as a consequence, it admits the usual product structure in an induced way. We prove further basic properties of H-clique-width in general.
Cite as

Petr Hliněný and Jan Jedelský. ℋ-Clique-Width and a Hereditary Analogue of Product Structure. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 61:1-61:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hlineny_et_al:LIPIcs.MFCS.2024.61,
  author =	{Hlin\v{e}n\'{y}, Petr and Jedelsk\'{y}, Jan},
  title =	{{ℋ-Clique-Width and a Hereditary Analogue of Product Structure}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{61:1--61:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.61},
  URN =		{urn:nbn:de:0030-drops-206176},
  doi =		{10.4230/LIPIcs.MFCS.2024.61},
  annote =	{Keywords: product structure, hereditary class, clique-width, twin-width}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2024.137
On Classes of Bounded Tree Rank, Their Interpretations, and Efficient Sparsification

Authors: Jakub Gajarský and Rose McCarty

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
Graph classes of bounded tree rank were introduced recently in the context of the model checking problem for first-order logic of graphs. These graph classes are a common generalization of graph classes of bounded degree and bounded treedepth, and they are a special case of graph classes of bounded expansion. We introduce a notion of decomposition for these classes and show that these decompositions can be efficiently computed. Also, a natural extension of our decomposition leads to a new characterization and decomposition for graph classes of bounded expansion (and an efficient algorithm computing this decomposition). We then focus on interpretations of graph classes of bounded tree rank. We give a characterization of graph classes interpretable in graph classes of tree rank 2. Importantly, our characterization leads to an efficient sparsification procedure: For any graph class 𝒞 interpretable in a graph class of tree rank at most 2, there is a polynomial time algorithm that to any G ∈ 𝒞 computes a (sparse) graph H from a fixed graph class of tree rank at most 2 such that G = I(H) for a fixed interpretation I. To the best of our knowledge, this is the first efficient "interpretation reversal" result that generalizes the result of Gajarský et al. [LICS 2016], who showed an analogous result for graph classes interpretable in classes of graphs of bounded degree.
Cite as

Jakub Gajarský and Rose McCarty. On Classes of Bounded Tree Rank, Their Interpretations, and Efficient Sparsification. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 137:1-137:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{gajarsky_et_al:LIPIcs.ICALP.2024.137,
  author =	{Gajarsk\'{y}, Jakub and McCarty, Rose},
  title =	{{On Classes of Bounded Tree Rank, Their Interpretations, and Efficient Sparsification}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{137:1--137:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.137},
  URN =		{urn:nbn:de:0030-drops-202802},
  doi =		{10.4230/LIPIcs.ICALP.2024.137},
  annote =	{Keywords: First-order model checking, structural graph theory, structural sparsity}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2023.11
Sparse Graphs of Twin-Width 2 Have Bounded Tree-Width

Authors: Benjamin Bergougnoux, Jakub Gajarský, Grzegorz Guśpiel, Petr Hliněný, Filip Pokrývka, and Marek Sokołowski

Published in: LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

Abstract
Twin-width is a structural width parameter introduced by Bonnet, Kim, Thomassé and Watrigant [FOCS 2020]. Very briefly, its essence is a gradual reduction (a contraction sequence) of the given graph down to a single vertex while maintaining limited difference of neighbourhoods of the vertices, and it can be seen as widely generalizing several other traditional structural parameters. Having such a sequence at hand allows to solve many otherwise hard problems efficiently. Our paper focuses on a comparison of twin-width to the more traditional tree-width on sparse graphs. Namely, we prove that if a graph G of twin-width at most 2 contains no K_{t,t} subgraph for some integer t, then the tree-width of G is bounded by a polynomial function of t. As a consequence, for any sparse graph class C we obtain a polynomial time algorithm which for any input graph G ∈ C either outputs a contraction sequence of width at most c (where c depends only on C), or correctly outputs that G has twin-width more than 2. On the other hand, we present an easy example of a graph class of twin-width 3 with unbounded tree-width, showing that our result cannot be extended to higher values of twin-width.
Cite as

Benjamin Bergougnoux, Jakub Gajarský, Grzegorz Guśpiel, Petr Hliněný, Filip Pokrývka, and Marek Sokołowski. Sparse Graphs of Twin-Width 2 Have Bounded Tree-Width. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 11:1-11:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bergougnoux_et_al:LIPIcs.ISAAC.2023.11,
  author =	{Bergougnoux, Benjamin and Gajarsk\'{y}, Jakub and Gu\'{s}piel, Grzegorz and Hlin\v{e}n\'{y}, Petr and Pokr\'{y}vka, Filip and Soko{\l}owski, Marek},
  title =	{{Sparse Graphs of Twin-Width 2 Have Bounded Tree-Width}},
  booktitle =	{34th International Symposium on Algorithms and Computation (ISAAC 2023)},
  pages =	{11:1--11:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-289-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{283},
  editor =	{Iwata, Satoru and Kakimura, Naonori},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.11},
  URN =		{urn:nbn:de:0030-drops-193130},
  doi =		{10.4230/LIPIcs.ISAAC.2023.11},
  annote =	{Keywords: twin-width, tree-width, excluded grid, sparsity}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2023.125
Indiscernibles and Flatness in Monadically Stable and Monadically NIP Classes

Authors: Jan Dreier, Nikolas Mählmann, Sebastian Siebertz, and Szymon Toruńczyk

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract
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 C of graphs is flip-flat if for every fixed radius r, every sufficiently large set of vertices of a graph G ∈ 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.
Cite as

Jan Dreier, Nikolas Mählmann, Sebastian Siebertz, and Szymon Toruńczyk. Indiscernibles and Flatness in Monadically Stable and Monadically NIP Classes. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 125:1-125:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{dreier_et_al:LIPIcs.ICALP.2023.125,
  author =	{Dreier, Jan and M\"{a}hlmann, Nikolas and Siebertz, Sebastian and Toru\'{n}czyk, Szymon},
  title =	{{Indiscernibles and Flatness in Monadically Stable and Monadically NIP Classes}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{125:1--125:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.125},
  URN =		{urn:nbn:de:0030-drops-181779},
  doi =		{10.4230/LIPIcs.ICALP.2023.125},
  annote =	{Keywords: stability, NIP, combinatorial characterization, first-order model checking}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2023.128
Flipper Games for Monadically Stable Graph Classes

Authors: Jakub Gajarský, Nikolas Mählmann, Rose McCarty, Pierre Ohlmann, Michał Pilipczuk, Wojciech Przybyszewski, Sebastian Siebertz, Marek Sokołowski, and Szymon Toruńczyk

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract
A class of graphs C is monadically stable if for every unary expansion Ĉ of C, one cannot encode - using first-order transductions - arbitrarily long linear orders in graphs from C. It is known that nowhere dense graph classes are monadically stable; these include classes of bounded maximum degree and classes that exclude a fixed topological minor. On the other hand, monadic stability is a property expressed in purely model-theoretic terms that is also suited for capturing structure in dense graphs. In this work we provide a characterization of monadic stability in terms of 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 Localizer, who in each round is forced to restrict 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 is based on tools borrowed 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 show that monadic stability for graph classes coincides with monadic stability of existential formulas with two free variables, and we provide another combinatorial characterization of monadic stability via forbidden patterns. The second proof relies on the recently introduced notion of flip-flatness (Dreier, Mählmann, Siebertz, and Toruńczyk, arXiv 2206.13765) and provides an efficient algorithm to compute Flipper’s moves in a winning strategy.
Cite as

Jakub Gajarský, Nikolas Mählmann, Rose McCarty, Pierre Ohlmann, Michał Pilipczuk, Wojciech Przybyszewski, Sebastian Siebertz, Marek Sokołowski, and Szymon Toruńczyk. Flipper Games for Monadically Stable Graph Classes. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 128:1-128:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{gajarsky_et_al:LIPIcs.ICALP.2023.128,
  author =	{Gajarsk\'{y}, Jakub and M\"{a}hlmann, Nikolas and McCarty, Rose and Ohlmann, Pierre and Pilipczuk, Micha{\l} and Przybyszewski, Wojciech and Siebertz, Sebastian and Soko{\l}owski, Marek and Toru\'{n}czyk, Szymon},
  title =	{{Flipper Games for Monadically Stable Graph Classes}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{128:1--128:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.128},
  URN =		{urn:nbn:de:0030-drops-181804},
  doi =		{10.4230/LIPIcs.ICALP.2023.128},
  annote =	{Keywords: Stability theory, structural graph theory, games}
}
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