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**Published in:** LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

We investigate a relaxation of the notion of treewidth-fragility, namely tree-independence-number-fragility. In particular, we obtain polynomial-time approximation schemes for independent packing problems on fractionally tree-independence-number-fragile graph classes. Our approach unifies and extends several known polynomial-time approximation schemes on seemingly unrelated graph classes, such as classes of intersection graphs of fat objects in a fixed dimension or proper minor-closed classes. We also study the related notion of layered tree-independence number, a relaxation of layered treewidth.

Esther Galby, Andrea Munaro, and Shizhou Yang. Polynomial-Time Approximation Schemes for Independent Packing Problems on Fractionally Tree-Independence-Number-Fragile Graphs. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 34:1-34:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{galby_et_al:LIPIcs.SoCG.2023.34, author = {Galby, Esther and Munaro, Andrea and Yang, Shizhou}, title = {{Polynomial-Time Approximation Schemes for Independent Packing Problems on Fractionally Tree-Independence-Number-Fragile Graphs}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {34:1--34:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.34}, URN = {urn:nbn:de:0030-drops-178840}, doi = {10.4230/LIPIcs.SoCG.2023.34}, annote = {Keywords: Independent packings, intersection graphs, polynomial-time approximation schemes, tree-independence number} }

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**Published in:** LIPIcs, Volume 249, 17th International Symposium on Parameterized and Exact Computation (IPEC 2022)

The leafage of a chordal graph G is the minimum integer 𝓁 such that G can be realized as an intersection graph of subtrees of a tree with 𝓁 leaves. We consider structural parameterization by the leafage of classical domination and cut problems on chordal graphs. Fomin, Golovach, and Raymond [ESA 2018, Algorithmica 2020] proved, among other things, that Dominating Set on chordal graphs admits an algorithm running in time 2^𝒪(𝓁²) ⋅ n^𝒪(1). We present a conceptually much simpler algorithm that runs in time 2^𝒪(𝓁) ⋅ n^𝒪(1). We extend our approach to obtain similar results for Connected Dominating Set and Steiner Tree. We then consider the two classical cut problems MultiCut with Undeletable Terminals and Multiway Cut with Undeletable Terminals. We prove that the former is W[1]-hard when parameterized by the leafage and complement this result by presenting a simple n^𝒪(𝓁)-time algorithm. To our surprise, we find that Multiway Cut with Undeletable Terminals on chordal graphs can be solved, in contrast, in n^O(1)-time.

Esther Galby, Dániel Marx, Philipp Schepper, Roohani Sharma, and Prafullkumar Tale. Domination and Cut Problems on Chordal Graphs with Bounded Leafage. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 14:1-14:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{galby_et_al:LIPIcs.IPEC.2022.14, author = {Galby, Esther and Marx, D\'{a}niel and Schepper, Philipp and Sharma, Roohani and Tale, Prafullkumar}, title = {{Domination and Cut Problems on Chordal Graphs with Bounded Leafage}}, booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)}, pages = {14:1--14:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-260-0}, ISSN = {1868-8969}, year = {2022}, volume = {249}, editor = {Dell, Holger and Nederlof, Jesper}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.14}, URN = {urn:nbn:de:0030-drops-173704}, doi = {10.4230/LIPIcs.IPEC.2022.14}, annote = {Keywords: Chordal Graphs, Leafage, FPT Algorithms, Dominating Set, MultiCut with Undeletable Terminals, Multiway Cut with Undeletable Terminals} }

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**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

For a graph G, a subset S ⊆ V(G) is called a resolving set if for any two vertices u,v ∈ V(G), there exists a vertex w ∈ S such that d(w,u) ≠ d(w,v). The Metric Dimension problem takes as input a graph G and a positive integer k, and asks whether there exists a resolving set of size at most k. This problem was introduced in the 1970s and is known to be NP-hard [GT 61 in Garey and Johnson’s book]. In the realm of parameterized complexity, Hartung and Nichterlein [CCC 2013] proved that the problem is W[2]-hard when parameterized by the natural parameter k. They also observed that it is FPT when parameterized by the vertex cover number and asked about its complexity under smaller parameters, in particular the feedback vertex set number. We answer this question by proving that Metric Dimension is W[1]-hard when parameterized by the feedback vertex set number. This also improves the result of Bonnet and Purohit [IPEC 2019] which states that the problem is W[1]-hard parameterized by the treewidth. Regarding the parameterization by the vertex cover number, we prove that Metric Dimension does not admit a polynomial kernel under this parameterization unless NP ⊆ coNP/poly. We observe that a similar result holds when the parameter is the distance to clique. On the positive side, we show that Metric Dimension is FPT when parameterized by either the distance to cluster or the distance to co-cluster, both of which are smaller parameters than the vertex cover number.

Esther Galby, Liana Khazaliya, Fionn Mc Inerney, Roohani Sharma, and Prafullkumar Tale. Metric Dimension Parameterized by Feedback Vertex Set and Other Structural Parameters. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 51:1-51:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{galby_et_al:LIPIcs.MFCS.2022.51, author = {Galby, Esther and Khazaliya, Liana and Mc Inerney, Fionn and Sharma, Roohani and Tale, Prafullkumar}, title = {{Metric Dimension Parameterized by Feedback Vertex Set and Other Structural Parameters}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {51:1--51:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.51}, URN = {urn:nbn:de:0030-drops-168496}, doi = {10.4230/LIPIcs.MFCS.2022.51}, annote = {Keywords: Metric Dimension, Parameterized Complexity, Feedback Vertex Set} }

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**Published in:** LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

In this paper, we consider the following problem: given a connected graph G, can we reduce the domination number of G by one by using only one edge contraction? We show that the problem is NP-hard when restricted to {P_6,P_4+P_2}-free graphs and that it is coNP-hard when restricted to subcubic claw-free graphs and 2P_3-free graphs. As a consequence, we are able to establish a complexity dichotomy for the problem on H-free graphs when H is connected.

Esther Galby, Paloma T. Lima, and Bernard Ries. Blocking Dominating Sets for H-Free Graphs via Edge Contractions. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 21:1-21:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{galby_et_al:LIPIcs.ISAAC.2019.21, author = {Galby, Esther and Lima, Paloma T. and Ries, Bernard}, title = {{Blocking Dominating Sets for H-Free Graphs via Edge Contractions}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {21:1--21:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.21}, URN = {urn:nbn:de:0030-drops-115171}, doi = {10.4230/LIPIcs.ISAAC.2019.21}, annote = {Keywords: domination number, blocker problem, H-free graphs} }

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**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

In this paper, we study the following problem: given a connected graph G, can we reduce the domination number of G by at least one using k edge contractions, for some fixed integer k >= 0? We show that for k <= 2, the problem is coNP-hard. We further prove that for k=1, the problem is W[1]-hard parameterized by the size of a minimum dominating set plus the mim-width of the input graph, and that it remains NP-hard when restricted to P_9-free graphs, bipartite graphs and {C_3,...,C_{l}}-free graphs for any l >= 3. Finally, we show that for any k >= 1, the problem is polynomial-time solvable for P_5-free graphs and that it can be solved in FPT-time and XP-time when parameterized by tree-width and mim-width, respectively.

Esther Galby, Paloma T. Lima, and Bernard Ries. Reducing the Domination Number of Graphs via Edge Contractions. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 41:1-41:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{galby_et_al:LIPIcs.MFCS.2019.41, author = {Galby, Esther and Lima, Paloma T. and Ries, Bernard}, title = {{Reducing the Domination Number of Graphs via Edge Contractions}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {41:1--41:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.41}, URN = {urn:nbn:de:0030-drops-109856}, doi = {10.4230/LIPIcs.MFCS.2019.41}, annote = {Keywords: domination number, blocker problem, graph classes} }

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

We investigate the Matrix Powering Positivity Problem, PosMatPow: given an m X m square integer matrix M, a linear function f: Z^{m X m} -> Z with integer coefficients, and a positive integer n (encoded in binary), determine whether f(M^n) \geq 0. We show that for fixed dimensions m of 2 and 3, this problem is decidable in polynomial time.

Esther Galby, Joël Ouaknine, and James Worrell. On Matrix Powering in Low Dimensions. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 329-340, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{galby_et_al:LIPIcs.STACS.2015.329, author = {Galby, Esther and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{On Matrix Powering in Low Dimensions}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {329--340}, 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.2015.329}, URN = {urn:nbn:de:0030-drops-49240}, doi = {10.4230/LIPIcs.STACS.2015.329}, annote = {Keywords: matrix powering, complexity, Baker's theorem} }

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