Document

**Published in:** LIPIcs, Volume 216, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022)

We introduce differential games for FO logic of graphs, a variant of Ehrenfeucht-Fraïssé games in which the game is played on only one graph and the moves of both players are restricted. We prove that these games are strong enough to capture essential information about graphs from graph classes which are interpretable in nowhere dense graph classes. This, together with the newly introduced notion of differential locality and the fact that the restriction of possible moves by the players makes it easy to decide the winner of the game in some cases, leads to a new approach to the FO model checking problem which can be used on various graph classes interpretable in classes of sparse graphs.

Jakub Gajarský, Maximilian Gorsky, and Stephan Kreutzer. Differential Games, Locality, and Model Checking for FO Logic of Graphs. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 22:1-22:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{gajarsky_et_al:LIPIcs.CSL.2022.22, author = {Gajarsk\'{y}, Jakub and Gorsky, Maximilian and Kreutzer, Stephan}, title = {{Differential Games, Locality, and Model Checking for FO Logic of Graphs}}, booktitle = {30th EACSL Annual Conference on Computer Science Logic (CSL 2022)}, pages = {22:1--22:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-218-1}, ISSN = {1868-8969}, year = {2022}, volume = {216}, editor = {Manea, Florin and Simpson, Alex}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2022.22}, URN = {urn:nbn:de:0030-drops-157426}, doi = {10.4230/LIPIcs.CSL.2022.22}, annote = {Keywords: FO model checking, locality, Gaifman’s theorem, EF games} }

Document

**Published in:** LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

Shrub-depth is a width measure of graphs which, roughly speaking, corresponds to the smallest depth of a tree into which a graph can be encoded. It can be thought of as a low-depth variant of clique-width (or rank-width), similarly as treedepth is a low-depth variant of treewidth. We present an fpt algorithm for computing decompositions of graphs of bounded shrub-depth. To the best of our knowledge, this is the first algorithm which computes the decomposition directly, without use of rank-width decompositions and FO or MSO logic.

Jakub Gajarský and Stephan Kreutzer. Computing Shrub-Depth Decompositions. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 56:1-56:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{gajarsky_et_al:LIPIcs.STACS.2020.56, author = {Gajarsk\'{y}, Jakub and Kreutzer, Stephan}, title = {{Computing Shrub-Depth Decompositions}}, booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, pages = {56:1--56:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-140-5}, ISSN = {1868-8969}, year = {2020}, volume = {154}, editor = {Paul, Christophe and Bl\"{a}ser, Markus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.56}, URN = {urn:nbn:de:0030-drops-119177}, doi = {10.4230/LIPIcs.STACS.2020.56}, annote = {Keywords: shrub-depth, tree-model, decomposition, fixed-parameter tractability} }

Document

**Published in:** LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

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.

Stephan Kreutzer, Irene Muzi, Patrice Ossona de Mendez, Roman Rabinovich, and Sebastian Siebertz. Algorithmic Properties of Sparse Digraphs. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 46:1-46:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kreutzer_et_al:LIPIcs.STACS.2019.46, author = {Kreutzer, Stephan and Muzi, Irene and Ossona de Mendez, Patrice and Rabinovich, Roman and Siebertz, Sebastian}, title = {{Algorithmic Properties of Sparse Digraphs}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {46:1--46:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.46}, URN = {urn:nbn:de:0030-drops-102859}, doi = {10.4230/LIPIcs.STACS.2019.46}, annote = {Keywords: Directed graphs, graph algorithms, parameterized complexity, approximation} }

Document

**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We show that for no surface except for the plane does monadic second-order logic (MSO) have a zero-one-law - and not even a convergence law - on the class of (connected) graphs embeddable on the surface. In addition we show that every rational in [0,1] is the limiting probability of some MSO formula. This strongly refutes a conjecture by Heinig et al. (2014) who proved a convergence law for planar graphs, and a zero-one law for connected planar graphs, and also identified the so-called gaps of [0,1]: the subintervals that are not limiting probabilities of any MSO formula. The proof relies on a combination of methods from structural graph theory, especially large face-width embeddings of graphs on surfaces, analytic combinatorics, and finite model theory, and several parts of the proof may be of independent interest. In particular, we identify precisely the properties that make the zero-one law work on planar graphs but fail for every other surface.

Albert Atserias, Stephan Kreutzer, and Marc Noy. On Zero-One and Convergence Laws for Graphs Embeddable on a Fixed Surface. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 116:1-116:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{atserias_et_al:LIPIcs.ICALP.2018.116, author = {Atserias, Albert and Kreutzer, Stephan and Noy, Marc}, title = {{On Zero-One and Convergence Laws for Graphs Embeddable on a Fixed Surface}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {116:1--116:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.116}, URN = {urn:nbn:de:0030-drops-91206}, doi = {10.4230/LIPIcs.ICALP.2018.116}, annote = {Keywords: topological graph theory, monadic second-order logic, random graphs, zero-one law, convergence law} }

Document

**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

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 interpretations 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 decompositions, replacing treedepth by its dense analogue called shrubdepth.

Jakub Gajarský, Stephan Kreutzer, Jaroslav Nesetril, Patrice Ossona de Mendez, Michal Pilipczuk, Sebastian Siebertz, and Szymon Torunczyk. First-Order Interpretations of Bounded Expansion Classes. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 126:1-126:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{gajarsky_et_al:LIPIcs.ICALP.2018.126, author = {Gajarsk\'{y}, Jakub and Kreutzer, Stephan and Nesetril, Jaroslav and Ossona de Mendez, Patrice and Pilipczuk, Michal and Siebertz, Sebastian and Torunczyk, Szymon}, title = {{First-Order Interpretations of Bounded Expansion Classes}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {126:1--126:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.126}, URN = {urn:nbn:de:0030-drops-91300}, doi = {10.4230/LIPIcs.ICALP.2018.126}, annote = {Keywords: Logical interpretations/transductions, structurally sparse graphs, bounded expansion} }

Document

Invited Talk

**Published in:** LIPIcs, Volume 82, 26th EACSL Annual Conference on Computer Science Logic (CSL 2017)

The model-checking problem for a logic LLL is the problem
of decidig for a given formula phi in LLL and structure
AA whether the formula is true in the structure, i.e. whether
AA models phi.
Model-checking for logics such as First-Order Logic (FO) or
Monadic Second-Order Logic (MSO) has been studied intensively
in the literature, especially in the context of algorithmic meta-theorems
within the framework of parameterized complexity. However, in the past the
focus of this line of research was model-checking on classes of
sparse graphs, e.g. planar graphs, graph classes excluding a
minor or classes which are nowhere dense. By now, the complexity of
first-order model-checking on sparse classes of graphs is completely
understood. Hence, current research now focusses mainly on classes of
dense graphs.
In this talk we will briefly review the known results on sparse classes
of graphs and explain the complete classification of classes of sparse
graphs on which first-order model-checking is tractable.
In the second part we will then focus on recent and ongoing research
analysing the complexity of first-order model-checking on classes of
dense graphs.

Stephan Kreutzer. Current Trends and New Perspectives for First-Order Model Checking (Invited Talk). In 26th EACSL Annual Conference on Computer Science Logic (CSL 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 82, pp. 4:1-4:5, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{kreutzer:LIPIcs.CSL.2017.4, author = {Kreutzer, Stephan}, title = {{Current Trends and New Perspectives for First-Order Model Checking}}, booktitle = {26th EACSL Annual Conference on Computer Science Logic (CSL 2017)}, pages = {4:1--4:5}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-045-3}, ISSN = {1868-8969}, year = {2017}, volume = {82}, editor = {Goranko, Valentin and Dam, Mads}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2017.4}, URN = {urn:nbn:de:0030-drops-77095}, doi = {10.4230/LIPIcs.CSL.2017.4}, annote = {Keywords: Finite Model Theory, Computational Model Theory, Algorithmic Meta Theorems, Model-Checking, Logical Approaches in Graph Theory} }

Document

**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

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.

Kord Eickmeyer, Archontia C. Giannopoulou, Stephan Kreutzer, O-joung Kwon, Michal Pilipczuk, Roman Rabinovich, and Sebastian Siebertz. Neighborhood Complexity and Kernelization for Nowhere Dense Classes of Graphs. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 63:1-63:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{eickmeyer_et_al:LIPIcs.ICALP.2017.63, author = {Eickmeyer, Kord and Giannopoulou, Archontia C. and Kreutzer, Stephan and Kwon, O-joung and Pilipczuk, Michal and Rabinovich, Roman and Siebertz, Sebastian}, title = {{Neighborhood Complexity and Kernelization for Nowhere Dense Classes of Graphs}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {63:1--63:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.63}, URN = {urn:nbn:de:0030-drops-74288}, doi = {10.4230/LIPIcs.ICALP.2017.63}, annote = {Keywords: Graph Structure Theory, Nowhere Dense Graphs, Parameterized Complexity, Kernelization, Dominating Set} }

Document

**Published in:** LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

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.

Stephan Kreutzer, Roman Rabinovich, Sebastian Siebertz, and Grischa Weberstädt. Structural Properties and Constant Factor-Approximation of Strong Distance-r Dominating Sets in Sparse Directed Graphs. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 48:1-48:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{kreutzer_et_al:LIPIcs.STACS.2017.48, author = {Kreutzer, Stephan and Rabinovich, Roman and Siebertz, Sebastian and Weberst\"{a}dt, Grischa}, title = {{Structural Properties and Constant Factor-Approximation of Strong Distance-r Dominating Sets in Sparse Directed Graphs}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {48:1--48:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-028-6}, ISSN = {1868-8969}, year = {2017}, volume = {66}, editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.48}, URN = {urn:nbn:de:0030-drops-69868}, doi = {10.4230/LIPIcs.STACS.2017.48}, annote = {Keywords: Directed Graph Structure Theory, Bounded Expansion, Generalised Colouring Numbers, Splitter Game, Approximation Algorithms, Dominating Set} }

Document

**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

We study the version of the k-disjoint paths problem where k demand pairs (s_1,t_1), ..., (s_k,t_k) are specified in the input and the paths in the solution are allowed to intersect, but such that no vertex is on more than c paths. We show that on directed acyclic graphs the problem is solvable in time n^{O(d)} if we allow congestion k-d for k paths. Furthermore, we show that, under a suitable complexity theoretic assumption, the problem cannot be solved in time f(k)n^{o(d*log(d))} for any computable function f.

Saeed Akhoondian Amiri, Stephan Kreutzer, Dániel Marx, and Roman Rabinovich. Routing with Congestion in Acyclic Digraphs. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 7:1-7:11, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{amiri_et_al:LIPIcs.MFCS.2016.7, author = {Amiri, Saeed Akhoondian and Kreutzer, Stephan and Marx, D\'{a}niel and Rabinovich, Roman}, title = {{Routing with Congestion in Acyclic Digraphs}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {7:1--7:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.7}, URN = {urn:nbn:de:0030-drops-64244}, doi = {10.4230/LIPIcs.MFCS.2016.7}, annote = {Keywords: algorithms, disjoint paths, congestion, acyclic digraphs, XP, W\lbrack1\rbrack-hard} }

Document

**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

The generalised colouring numbers adm_r(G), col_r(G), and 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 Nesetril 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.

Stephan Kreutzer, Michal Pilipczuk, Roman Rabinovich, and Sebastian Siebertz. The Generalised Colouring Numbers on Classes of Bounded Expansion. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 85:1-85:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kreutzer_et_al:LIPIcs.MFCS.2016.85, author = {Kreutzer, Stephan and Pilipczuk, Michal and Rabinovich, Roman and Siebertz, Sebastian}, title = {{The Generalised Colouring Numbers on Classes of Bounded Expansion}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {85:1--85:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.85}, URN = {urn:nbn:de:0030-drops-64937}, doi = {10.4230/LIPIcs.MFCS.2016.85}, annote = {Keywords: graph structure theory, nowhere dense graphs, generalised colouring numbers, splitter game, first-order model-checking} }

Document

**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

We prove that for every positive integer r and for every graph class G of bounded expansion, the r-DOMINATING SET problem admits a linear kernel on graphs from G. Moreover, in the more general case when G is only assumed to be nowhere dense, we give an almost linear kernel on G for the classic DOMINATING SET problem, i.e., for the case r=1. These results generalize a line of previous research on finding linear kernels for DOMINATING SET and r-DOMINATING SET (Alber et al., JACM 2004, Bodlaender et al., FOCS 2009, Fomin et al., SODA 2010, Fomin et al., SODA 2012, Fomin et al., STACS 2013). However, the approach taken in this work, which is based on the theory of sparse graphs, is radically different and conceptually much simpler than the previous approaches.
We complement our findings by showing that for the closely related CONNECTED DOMINATING SET problem, the existence of such kernelization algorithms is unlikely, even though the problem is known to admit a linear kernel on H-topological-minor-free graphs (Fomin et al., STACS 2013). Also, we prove that for any somewhere dense class G, there is some r for which r-DOMINATING SET is W[2]-hard on G. Thus, our results fall short of proving a sharp dichotomy for the parameterized complexity of r-DOMINATING SET on subgraph-monotone graph classes: we conjecture that the border of tractability lies exactly between nowhere dense and somewhere dense graph classes.

Pål Grønås Drange, Markus Dregi, Fedor V. Fomin, Stephan Kreutzer, Daniel Lokshtanov, Marcin Pilipczuk, Michal Pilipczuk, Felix Reidl, Fernando Sánchez Villaamil, Saket Saurabh, Sebastian Siebertz, and Somnath Sikdar. Kernelization and Sparseness: the Case of Dominating Set. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 31:1-31:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{drange_et_al:LIPIcs.STACS.2016.31, author = {Drange, P\r{a}l Gr{\o}n\r{a}s and Dregi, Markus and Fomin, Fedor V. and Kreutzer, Stephan and Lokshtanov, Daniel and Pilipczuk, Marcin and Pilipczuk, Michal and Reidl, Felix and S\'{a}nchez Villaamil, Fernando and Saurabh, Saket and Siebertz, Sebastian and Sikdar, Somnath}, title = {{Kernelization and Sparseness: the Case of Dominating Set}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {31:1--31:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.31}, URN = {urn:nbn:de:0030-drops-57327}, doi = {10.4230/LIPIcs.STACS.2016.31}, annote = {Keywords: kernelization, dominating set, bounded expansion, nowhere dense} }

Document

Complete Volume

**Published in:** LIPIcs, Volume 41, 24th EACSL Annual Conference on Computer Science Logic (CSL 2015)

LIPIcs, Volume 41, CSL'15, Complete Volume

Stephan Kreutzer. LIPIcs, Volume 41, CSL'15, Complete Volume. In 24th EACSL Annual Conference on Computer Science Logic (CSL 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 41, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)

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@Proceedings{kreutzer:LIPIcs.CSL.2015, title = {{LIPIcs, Volume 41, CSL'15, Complete Volume}}, booktitle = {24th EACSL Annual Conference on Computer Science Logic (CSL 2015)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-90-3}, ISSN = {1868-8969}, year = {2015}, volume = {41}, editor = {Kreutzer, Stephan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2015}, URN = {urn:nbn:de:0030-drops-54670}, doi = {10.4230/LIPIcs.CSL.2015}, annote = {Keywords: Conference Proceedings, Distributed Systems, Software/ Programs Verifications, Formal Definitions and Theory, Languages Constructs and Features, Knowledge Representations Formalisms and Methods, Theory of Computation, Mathematical Logic} }

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Front Matter

**Published in:** LIPIcs, Volume 41, 24th EACSL Annual Conference on Computer Science Logic (CSL 2015)

Front Matter, Table of Contents, Preface, Conference Organisation, External Reviewers

Stephan Kreutzer. Front Matter, Table of Contents, Preface, Conference Organisation, External Reviewers. In 24th EACSL Annual Conference on Computer Science Logic (CSL 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 41, pp. i-xiv, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)

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@InProceedings{kreutzer:LIPIcs.CSL.2015.i, author = {Kreutzer, Stephan}, title = {{Front Matter, Table of Contents, Preface, Conference Organisation, External Reviewers}}, booktitle = {24th EACSL Annual Conference on Computer Science Logic (CSL 2015)}, pages = {i--xiv}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-90-3}, ISSN = {1868-8969}, year = {2015}, volume = {41}, editor = {Kreutzer, Stephan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2015.i}, URN = {urn:nbn:de:0030-drops-54489}, doi = {10.4230/LIPIcs.CSL.2015.i}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organisation, External Reviewers} }

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**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

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.

Saeed Akhoondian Amiri, Lukasz Kaiser, Stephan Kreutzer, Roman Rabinovich, and Sebastian Siebertz. Graph Searching Games and Width Measures for Directed Graphs. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 34-47, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)

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@InProceedings{akhoondianamiri_et_al:LIPIcs.STACS.2015.34, author = {Akhoondian Amiri, Saeed and Kaiser, Lukasz and Kreutzer, Stephan and Rabinovich, Roman and Siebertz, Sebastian}, title = {{Graph Searching Games and Width Measures for Directed Graphs}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {34--47}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.34}, URN = {urn:nbn:de:0030-drops-49020}, doi = {10.4230/LIPIcs.STACS.2015.34}, annote = {Keywords: cops and robber games, directed graphs, DAG-width} }

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Invited Talk

**Published in:** LIPIcs, Volume 24, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)

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.

Martin Grohe, Stephan Kreutzer, and Sebastian Siebertz. Characterisations of Nowhere Dense Graphs (Invited Talk). In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 24, pp. 21-40, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)

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@InProceedings{grohe_et_al:LIPIcs.FSTTCS.2013.21, author = {Grohe, Martin and Kreutzer, Stephan and Siebertz, Sebastian}, title = {{Characterisations of Nowhere Dense Graphs}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)}, pages = {21--40}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-64-4}, ISSN = {1868-8969}, year = {2013}, volume = {24}, editor = {Seth, Anil and Vishnoi, Nisheeth K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2013.21}, URN = {urn:nbn:de:0030-drops-44029}, doi = {10.4230/LIPIcs.FSTTCS.2013.21}, annote = {Keywords: Graph Algorithms, Algorithmic Graph Structure Theory, Finite Model Theory, Nowhere Dense Classes of Graphs} }

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**Published in:** Dagstuhl Reports, Volume 1, Issue 2 (2011)

From February 14, 2012 to February 18, 2012, the Dagstuhl Seminar 11071
``Theory and Applications of Graph Searching Problems (GRASTA 2011)''
was held in Schloss Dagstuhl~--~Leibniz Center for Informatics.
During the seminar, participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and open problems are put together in this paper. The first section describes the seminar topics and goals in general.
The second section contains the abstracts of the talks and the third section
includes the open problems presented during the seminar.

Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer, and Dimitrios M. Thilikos. Theory and Applications of Graph Searching Problems (GRASTA 2011) (Dagstuhl Seminar 11071). In Dagstuhl Reports, Volume 1, Issue 2, pp. 30-46, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2011)

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@Article{fomin_et_al:DagRep.1.2.30, author = {Fomin, Fedor V. and Fraigniaud, Pierre and Kreutzer, Stephan and Thilikos, Dimitrios M.}, title = {{Theory and Applications of Graph Searching Problems (GRASTA 2011) (Dagstuhl Seminar 11071)}}, pages = {30--46}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2011}, volume = {1}, number = {2}, editor = {Fomin, Fedor V. and Fraigniaud, Pierre and Kreutzer, Stephan and Thilikos, Dimitrios M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.1.2.30}, URN = {urn:nbn:de:0030-drops-31534}, doi = {10.4230/DagRep.1.2.30}, annote = {Keywords: Graph Searching, Pursuit Evasion Games, Cop and Robers Games, Fugitive Search Games} }

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**Published in:** LIPIcs, Volume 4, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2009)

We investigate the parameterized complexity of generalisations and variations
of the dominating set problem on classes of graphs that are nowhere dense. In
particular, we show that the distance-$d$ dominating-set problem, also known
as the $(k,d)$-centres problem, is fixed-parameter tractable on any class that
is nowhere dense and closed under induced subgraphs. This generalises known
results about the dominating set problem on $H$-minor free classes, classes
with locally excluded minors and classes of graphs of bounded expansion. A
key feature of our proof is that it is based simply on the fact that these
graph classes are uniformly quasi-wide, and does not rely on a structural
decomposition. Our result also establishes that the distance-$d$
dominating-set problem is FPT on classes of bounded expansion, answering a
question of Ne{\v s}et{\v r}il and Ossona de Mendez.

Anuj Dawar and Stephan Kreutzer. Domination Problems in Nowhere-Dense Classes. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 157-168, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2009)

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@InProceedings{dawar_et_al:LIPIcs.FSTTCS.2009.2315, author = {Dawar, Anuj and Kreutzer, Stephan}, title = {{Domination Problems in Nowhere-Dense Classes}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science}, pages = {157--168}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-13-2}, ISSN = {1868-8969}, year = {2009}, volume = {4}, editor = {Kannan, Ravi and Narayan Kumar, K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2009.2315}, URN = {urn:nbn:de:0030-drops-23153}, doi = {10.4230/LIPIcs.FSTTCS.2009.2315}, annote = {Keywords: Dominating Set, distance-d dominating set, nowhere-dense graph classes, H-minor free graphs, fixed-parameter tractablility} }

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