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

We study a variation of the cops and robber game characterising treewidth, where in each round at most one cop may be placed and in each play at most q rounds are played, where q is a parameter of the game. We prove that if k cops have a winning strategy in this game, then k cops have a monotone winning strategy. As a corollary we obtain a new characterisation of bounded depth treewidth, and we give a positive answer to an open question by Fluck, Seppelt and Spitzer (2024), thus showing that graph classes of bounded depth treewidth are homomorphism distinguishing closed.
Our proof of monotonicity substantially reorganises a winning strategy by first transforming it into a pre-tree decomposition, which is inspired by decompositions of matroids, and then applying an intricate breadth-first "cleaning up" procedure along the pre-tree decomposition (which may temporarily lose the property of representing a strategy), in order to achieve monotonicity while controlling the number of rounds simultaneously across all branches of the decomposition via a vertex exchange argument. We believe this can be useful in future research.

Isolde Adler and Eva Fluck. Monotonicity of the Cops and Robber Game for Bounded Depth Treewidth. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{adler_et_al:LIPIcs.MFCS.2024.6, author = {Adler, Isolde and Fluck, Eva}, title = {{Monotonicity of the Cops and Robber Game for Bounded Depth Treewidth}}, booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)}, pages = {6:1--6:18}, 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.2024.6}, URN = {urn:nbn:de:0030-drops-205621}, doi = {10.4230/LIPIcs.MFCS.2024.6}, annote = {Keywords: tree decompositions, treewidth, treedepth, cops-and-robber game, monotonicity, homomorphism distinguishing closure} }

Document

**Published in:** LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

In Property Testing, proximity-oblivious testers (POTs) form a class of particularly simple testing algorithms, where a basic test is performed a number of times that may depend on the proximity parameter, but the basic test itself is independent of the proximity parameter.
In their seminal work, Goldreich and Ron [STOC 2009; SICOMP 2011] show that the graph properties that allow constant-query proximity-oblivious testing in the bounded-degree model are precisely the properties that can be expressed as a generalised subgraph freeness (GSF) property that satisfies the non-propagation condition. It is left open whether the non-propagation condition is necessary. Indeed, calling properties expressible as a generalised subgraph freeness property GSF-local properties, they ask whether all GSF-local properties are non-propagating. We give a negative answer by exhibiting a property of graphs that is GSF-local and propagating. Hence in particular, our property does not admit a POT, despite being GSF-local. We prove our result by exploiting a recent work of the authors which constructed a first-order (FO) property that is not testable [SODA 2021], and a new connection between FO properties and GSF-local properties via neighbourhood profiles.

Isolde Adler, Noleen Köhler, and Pan Peng. GSF-Locality Is Not Sufficient For Proximity-Oblivious Testing. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 34:1-34:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{adler_et_al:LIPIcs.CCC.2021.34, author = {Adler, Isolde and K\"{o}hler, Noleen and Peng, Pan}, title = {{GSF-Locality Is Not Sufficient For Proximity-Oblivious Testing}}, booktitle = {36th Computational Complexity Conference (CCC 2021)}, pages = {34:1--34:27}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-193-1}, ISSN = {1868-8969}, year = {2021}, volume = {200}, editor = {Kabanets, Valentine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.34}, URN = {urn:nbn:de:0030-drops-143082}, doi = {10.4230/LIPIcs.CCC.2021.34}, annote = {Keywords: Property testing, proximity-oblivous testing, locality, first-order logic, lower bound} }

Document

**Published in:** LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

Property testing algorithms are highly efficient algorithms, that come with probabilistic accuracy guarantees. For a property P, the goal is to distinguish inputs that have P from those that are far from having P with high probability correctly, by querying only a small number of local parts of the input. In property testing on graphs, the distance is measured by the number of edge modifications (additions or deletions), that are necessary to transform a graph into one with property P. Much research has focussed on the query complexity of such algorithms, i. e. the number of queries the algorithm makes to the input, but in view of applications, the running time of the algorithm is equally relevant.
In (Adler, Harwath STACS 2018), a natural extension of the bounded degree graph model of property testing to relational databases of bounded degree was introduced, and it was shown that on databases of bounded degree and bounded tree-width, every property that is expressible in monadic second-order logic with counting (CMSO) is testable with constant query complexity and sublinear running time. It remains open whether this can be improved to constant running time.
In this paper we introduce a new model, which is based on the bounded degree model, but the distance measure allows both edge (tuple) modifications and vertex (element) modifications. Our main theorem shows that on databases of bounded degree and bounded tree-width, every property that is expressible in CMSO is testable with constant query complexity and constant running time in the new model. We also show that every property that is testable in the classical model is testable in our model with the same query complexity and running time, but the converse is not true.
We argue that our model is natural and our meta-theorem showing constant-time CMSO testability supports this.

Isolde Adler and Polly Fahey. Faster Property Testers in a Variation of the Bounded Degree Model. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{adler_et_al:LIPIcs.FSTTCS.2020.7, author = {Adler, Isolde and Fahey, Polly}, title = {{Faster Property Testers in a Variation of the Bounded Degree Model}}, booktitle = {40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)}, pages = {7:1--7:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-174-0}, ISSN = {1868-8969}, year = {2020}, volume = {182}, editor = {Saxena, Nitin and Simon, Sunil}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.7}, URN = {urn:nbn:de:0030-drops-132482}, doi = {10.4230/LIPIcs.FSTTCS.2020.7}, annote = {Keywords: Constant Time Algorithms, Logic and Databases, Property Testing, Bounded Degree Model} }

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**Published in:** LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

The node search game against a lazy (or, respectively, agile) invisible robber has been introduced as a search-game analogue of the treewidth parameter (and, respectively, pathwidth). In the connected variants of the above two games, we additionally demand that, at each moment of the search, the clean territories are connected. The connected search game against an agile and invisible robber has been extensively examined. The monotone variant (where we also demand that the clean territories are progressively increasing) of this game, corresponds to the graph parameter of connected pathwidth. It is known that the price of connectivty to search for an agile robber is bounded by 2, that is the connected pathwidth of a graph is at most twice (plus some constant) its pathwidth. In this paper, we investigate the connected search game against a lazy robber. A lazy robber moves only when the searchers' strategy threatens the location that he currently occupies. We introduce two alternative graph-theoretic formulations of this game, one in terms of connected tree decompositions, and one in terms of (connected) layouts, leading to the graph parameter of connected treewidth. We observe that connected treewidth parameter is closed under contractions and prove that for every k >= 2, the set of contraction obstructions of the class of graphs with connected treewidth at most k is infinite. Our main result is a complete characterization of the obstruction set for k=2. One may observe that, so far, only a few complete obstruction sets are explicitly known for contraction closed graph classes. We finally show that, in contrast to the agile robber game, the price of connectivity is unbounded.

Isolde Adler, Christophe Paul, and Dimitrios M. Thilikos. Connected Search for a Lazy Robber. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 7:1-7:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{adler_et_al:LIPIcs.FSTTCS.2019.7, author = {Adler, Isolde and Paul, Christophe and Thilikos, Dimitrios M.}, title = {{Connected Search for a Lazy Robber}}, booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)}, pages = {7:1--7:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-131-3}, ISSN = {1868-8969}, year = {2019}, volume = {150}, editor = {Chattopadhyay, Arkadev and Gastin, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.7}, URN = {urn:nbn:de:0030-drops-115695}, doi = {10.4230/LIPIcs.FSTTCS.2019.7}, annote = {Keywords: Graph searching, Tree-decomposition, Obstructions} }

Document

**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

Aiming at extremely efficient algorithms for big data sets, we introduce property testing of relational databases of bounded degree. Our model generalises the bounded degree model for graphs (Goldreich and Ron, STOC 1997). We prove that in this model, if the databases have bounded tree-width, then every query definable in monadic second-order logic with modulo counting is testable with a constant number of oracle queries and polylogarithmic running time. This is the first logical meta-theorem in property testing of sparse models. Furthermore, we discuss conditions for the existence of uniform and non-uniform testers.

Isolde Adler and Frederik Harwath. Property Testing for Bounded Degree Databases. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{adler_et_al:LIPIcs.STACS.2018.6, author = {Adler, Isolde and Harwath, Frederik}, title = {{Property Testing for Bounded Degree Databases}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {6:1--6:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf 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.2018.6}, URN = {urn:nbn:de:0030-drops-85288}, doi = {10.4230/LIPIcs.STACS.2018.6}, annote = {Keywords: logic and databases, property testing, logical meta-theorems, bounded degree model, sublinear algorithms} }

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