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

The goal of local certification is to locally convince the vertices of a graph G that G satisfies a given property. A prover assigns short certificates to the vertices of the graph, then the vertices are allowed to check their certificates and the certificates of their neighbors, and based only on this local view and their own unique identifier, they must decide whether G satisfies the given property. If the graph indeed satisfies the property, all vertices must accept the instance, and otherwise at least one vertex must reject the instance (for any possible assignment of certificates). The goal is to minimize the size of the certificates.
In this paper we study the local certification of geometric and topological graph classes. While it is known that in n-vertex graphs, planarity can be certified locally with certificates of size O(log n), we show that several closely related graph classes require certificates of size Ω(n). This includes penny graphs, unit-distance graphs, (induced) subgraphs of the square grid, 1-planar graphs, and unit-square graphs. These bounds are tight up to a constant factor and give the first known examples of hereditary (and even monotone) graph classes for which the certificates must have linear size. For unit-disk graphs we obtain a lower bound of Ω(n^{1-δ}) for any δ > 0 on the size of the certificates, and an upper bound of O(n log n). The lower bounds are obtained by proving rigidity properties of the considered graphs, which might be of independent interest.

Oscar Defrain, Louis Esperet, Aurélie Lagoutte, Pat Morin, and Jean-Florent Raymond. Local Certification of Geometric Graph Classes. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 48:1-48:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{defrain_et_al:LIPIcs.MFCS.2024.48, author = {Defrain, Oscar and Esperet, Louis and Lagoutte, Aur\'{e}lie and Morin, Pat and Raymond, Jean-Florent}, title = {{Local Certification of Geometric Graph Classes}}, booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)}, pages = {48:1--48:14}, 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.2024.48}, URN = {urn:nbn:de:0030-drops-206042}, doi = {10.4230/LIPIcs.MFCS.2024.48}, annote = {Keywords: Local certification, proof labeling schemes, geometric intersection graphs} }

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**Published in:** LIPIcs, Volume 214, 16th International Symposium on Parameterized and Exact Computation (IPEC 2021)

At IPEC 2020, Bergougnoux, Bonnet, Brettell, and Kwon (Close Relatives of Feedback Vertex Set Without Single-Exponential Algorithms Parameterized by Treewidth, IPEC 2020, LIPIcs vol. 180, pp. 3:1-3:17) showed that a number of problems related to the classic Feedback Vertex Set (FVS) problem do not admit a 2^{o(k log k)} ⋅ n^{𝒪(1)}-time algorithm on graphs of treewidth at most k, assuming the Exponential Time Hypothesis. This contrasts with the 3^{k} ⋅ k^{𝒪(1)} ⋅ n-time algorithm for FVS using the Cut&Count technique.
During their live talk at IPEC 2020, Bergougnoux et al. posed a number of open questions, which we answer in this work.
- Subset Even Cycle Transversal, Subset Odd Cycle Transversal, Subset Feedback Vertex Set can be solved in time 2^{𝒪(k log k)} ⋅ n in graphs of treewidth at most k. This matches a lower bound for Even Cycle Transversal of Bergougnoux et al. and improves the polynomial factor in some of their upper bounds.
- Subset Feedback Vertex Set and Node Multiway Cut can be solved in time 2^{𝒪(k log k)} ⋅ n, if the input graph is given as a cliquewidth expression of size n and width k.
- Odd Cycle Transversal can be solved in time 4^k ⋅ k^{𝒪(1)} ⋅ n if the input graph is given as a cliquewidth expression of size n and width k. Furthermore, the existence of a constant ε > 0 and an algorithm performing this task in time (4-ε)^k ⋅ n^{𝒪(1)} would contradict the Strong Exponential Time Hypothesis. A common theme of the first two algorithmic results is to represent connectivity properties of the current graph in a state of a dynamic programming algorithm as an auxiliary forest with 𝒪(k) nodes. This results in a 2^{𝒪(k log k)} bound on the number of states for one node of the tree decomposition or cliquewidth expression and allows to compare two states in k^{𝒪(1)} time, resulting in linear time dependency on the size of the graph or the input cliquewidth expression.

Hugo Jacob, Thomas Bellitto, Oscar Defrain, and Marcin Pilipczuk. Close Relatives (Of Feedback Vertex Set), Revisited. In 16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{jacob_et_al:LIPIcs.IPEC.2021.21, author = {Jacob, Hugo and Bellitto, Thomas and Defrain, Oscar and Pilipczuk, Marcin}, title = {{Close Relatives (Of Feedback Vertex Set), Revisited}}, booktitle = {16th International Symposium on Parameterized and Exact Computation (IPEC 2021)}, pages = {21:1--21:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-216-7}, ISSN = {1868-8969}, year = {2021}, volume = {214}, editor = {Golovach, Petr A. and Zehavi, Meirav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2021.21}, URN = {urn:nbn:de:0030-drops-154049}, doi = {10.4230/LIPIcs.IPEC.2021.21}, annote = {Keywords: feedback vertex set, treewidth, cliquewidth} }

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

In [M. M. Kanté, V. Limouzy, A. Mary, and L. Nourine. On the enumeration of minimal dominating sets and related notions. SIAM Journal on Discrete Mathematics, 28(4):1916–1929, 2014.] the authors give an O(n+m) delay algorithm based on neighborhood inclusions for the enumeration of minimal dominating sets in split and P_6-free chordal graphs. In this paper, we investigate generalizations of this technique to P_k-free chordal graphs for larger integers k. In particular, we give O(n+m) and O(n^3 * m) delays algorithms in the classes of P_7-free and P_8-free chordal graphs. As for P_k-free chordal graphs for k >= 9, we give evidence that such a technique is inefficient as a key step of the algorithm, namely the irredundant extension problem, becomes NP-complete.

Oscar Defrain and Lhouari Nourine. Neighborhood Inclusions for Minimal Dominating Sets Enumeration: Linear and Polynomial Delay Algorithms in P_7 - Free and P_8 - Free Chordal Graphs. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 63:1-63:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{defrain_et_al:LIPIcs.ISAAC.2019.63, author = {Defrain, Oscar and Nourine, Lhouari}, title = {{Neighborhood Inclusions for Minimal Dominating Sets Enumeration: Linear and Polynomial Delay Algorithms in P\underline7 - Free and P\underline8 - Free Chordal Graphs}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {63:1--63:16}, 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.2019.63}, URN = {urn:nbn:de:0030-drops-115591}, doi = {10.4230/LIPIcs.ISAAC.2019.63}, annote = {Keywords: Minimal dominating sets, enumeration algorithms, linear delay enumeration, chordal graphs, forbidden induced paths} }

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

It is a long-standing open problem whether the minimal dominating sets of a graph can be enumerated in output-polynomial time. In this paper we prove that this is the case in triangle-free graphs. This answers a question of Kanté et al. Additionally, we show that deciding if a set of vertices of a bipartite graph can be completed into a minimal dominating set is a NP-complete problem.

Marthe Bonamy, Oscar Defrain, Marc Heinrich, and Jean-Florent Raymond. Enumerating Minimal Dominating Sets in Triangle-Free Graphs. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 16:1-16:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bonamy_et_al:LIPIcs.STACS.2019.16, author = {Bonamy, Marthe and Defrain, Oscar and Heinrich, Marc and Raymond, Jean-Florent}, title = {{Enumerating Minimal Dominating Sets in Triangle-Free Graphs}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {16:1--16:12}, 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.2019.16}, URN = {urn:nbn:de:0030-drops-102557}, doi = {10.4230/LIPIcs.STACS.2019.16}, annote = {Keywords: Enumeration algorithms, output-polynomial algorithms, minimal dominating set, triangle-free graphs, split graphs} }

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