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

The notion of tree-cut width has been introduced by Wollan in [The structure of graphs not admitting a fixed immersion, Journal of Combinatorial Theory, Series B, 110:47 - 66, 2015]. It is defined via tree-cut decompositions, which are tree-like decompositions that highlight small (edge) cuts in a graph. In that sense, tree-cut decompositions can be seen as an edge-version of tree-decompositions and have algorithmic applications on problems that remain intractable on graphs of bounded treewidth. In this paper, we prove that every graph admits an optimal tree-cut decomposition that satisfies a certain Menger-like condition similar to that of the lean tree decompositions of Thomas [A Menger-like property of tree-width: The finite case, Journal of Combinatorial Theory, Series B, 48(1):67 - 76, 1990]. This allows us to give, for every k in N, an upper-bound on the number immersion-minimal graphs of tree-cut width k. Our results imply the constructive existence of a linear FPT-algorithm for tree-cut width.

Archontia C. Giannopoulou, O-joung Kwon, Jean-Florent Raymond, and Dimitrios M. Thilikos. Lean Tree-Cut Decompositions: Obstructions and Algorithms. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 32:1-32:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{giannopoulou_et_al:LIPIcs.STACS.2019.32, author = {Giannopoulou, Archontia C. and Kwon, O-joung and Raymond, Jean-Florent and Thilikos, Dimitrios M.}, title = {{Lean Tree-Cut Decompositions: Obstructions and Algorithms}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {32:1--32:14}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.32}, URN = {urn:nbn:de:0030-drops-102716}, doi = {10.4230/LIPIcs.STACS.2019.32}, annote = {Keywords: tree-cut width, lean decompositions, immersions, obstructions, parameterized algorithms} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

For a graph H, a graph G is an H-graph if it is an intersection graph of connected subgraphs of some subdivision of H. These graphs naturally generalize several important graph classes like interval graphs or circular-arc graph. This notion was introduced in the early 1990s by Biro, Hujter, and Tuza. Recently, Chaplick et al. initiated the algorithmic study of H-graphs by showing that a number of fundamental optimization problems like Clique, Independent Set, or Dominating Set are solvable in polynomial time on H-graphs. We extend and complement these algorithmic findings in several directions.
First we show that for every fixed H, the class of H-graphs is of logarithmically-bounded boolean-width. We also prove that H-graphs are graphs with polynomially many minimal separators. Pipelined with the plethora of known algorithms on graphs of bounded boolean-width and graphs with polynomially many minimal separators, this describes a large class of optimization problems that are solvable in polynomial time on H-graphs.
The most fundamental optimization problems among those solvable in polynomial time on H-graphs are Clique, Independent Set, and Dominating Set. We provide a more refined complexity analysis of these problems from the perspective of parameterized complexity. We show that Independent Set and Dominating Set are W[1]-hard being parameterized by the size of H plus the size of the solution. On the other hand, we prove that when H is a tree, Dominating Set is fixed-parameter tractable (FPT) parameterized by the size of H. Besides, we show that Clique admits a polynomial kernel parameterized by H and the solution size.

Fedor V. Fomin, Petr A. Golovach, and Jean-Florent Raymond. On the Tractability of Optimization Problems on H-Graphs. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 30:1-30:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{fomin_et_al:LIPIcs.ESA.2018.30, author = {Fomin, Fedor V. and Golovach, Petr A. and Raymond, Jean-Florent}, title = {{On the Tractability of Optimization Problems on H-Graphs}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {30:1--30:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.30}, URN = {urn:nbn:de:0030-drops-94930}, doi = {10.4230/LIPIcs.ESA.2018.30}, annote = {Keywords: H-topological intersection graphs, parameterized complexity, minimal separators, boolean-width, mim-width} }

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

Suppose F is a finite family of graphs. We consider the following meta-problem, called F-Immersion Deletion: given a graph G and an integer k, decide whether the deletion of at most k edges of G can result in a graph that does not contain any graph from F as an immersion. This problem is a close relative of the F-Minor Deletion problem studied by Fomin et al. [FOCS 2012], where one deletes vertices in order to remove all minor models of graphs from F.
We prove that whenever all graphs from F are connected and at least one graph of F is planar and subcubic, then the F-Immersion Deletion problem admits:
- a constant-factor approximation algorithm running in time O(m^3 n^3 log m)
- a linear kernel that can be computed in time O(m^4 n^3 log m) and
- a O(2^{O(k)} + m^4 n^3 log m)-time fixed-parameter algorithm,
where n,m count the vertices and edges of the input graph. Our findings mirror those of Fomin et al. [FOCS 2012], who obtained similar results for F-Minor Deletion, under the assumption that at least one graph from F is planar.
An important difference is that we are able to obtain a linear kernel for F-Immersion Deletion, while the exponent of the kernel of Fomin et al. depends heavily on the family F. In fact, this dependence is unavoidable under plausible complexity assumptions, as proven by Giannopoulou et al. [ICALP 2015]. This reveals that the kernelization complexity of F-Immersion Deletion is quite different than that of F-Minor Deletion.

Archontia C. Giannopoulou, Michal Pilipczuk, Jean-Florent Raymond, Dimitrios M. Thilikos, and Marcin Wrochna. Linear Kernels for Edge Deletion Problems to Immersion-Closed Graph Classes. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 57:1-57:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{giannopoulou_et_al:LIPIcs.ICALP.2017.57, author = {Giannopoulou, Archontia C. and Pilipczuk, Michal and Raymond, Jean-Florent and Thilikos, Dimitrios M. and Wrochna, Marcin}, title = {{Linear Kernels for Edge Deletion Problems to Immersion-Closed Graph Classes}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {57:1--57:15}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.57}, URN = {urn:nbn:de:0030-drops-73891}, doi = {10.4230/LIPIcs.ICALP.2017.57}, annote = {Keywords: Kernelization, Approximation, Immersion, Protrusion, Tree-cut width} }

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**Published in:** LIPIcs, Volume 63, 11th International Symposium on Parameterized and Exact Computation (IPEC 2016)

Cutwidth is one of the classic layout parameters for graphs. It measures how well one can order the vertices of a graph in a linear manner, so that the maximum number of edges between any prefix and its complement suffix is minimized. As graphs of cutwidth at most k are closed under taking immersions, the results of Robertson and Seymour imply that there is a finite list of minimal immersion obstructions for admitting a cut layout of width at most k. We prove that every minimal immersion obstruction for cutwidth at most k has size at most 2^O(k^3*log(k)).
As an interesting algorithmic byproduct, we design a new fixed-parameter algorithm for computing the cutwidth of a graph that runs in time 2^O(k^2*log(k))*n, where k is the optimum width and n is the number of vertices. While being slower by a log k-factor in the exponent than the fastest known algorithm, due to Thilikos, Bodlaender, and Serna [J. Algorithms 2005], our algorithm has the advantage of being simpler and self-contained; arguably, it explains better the combinatorics of optimum-width layouts.

Archontia C. Giannopoulou, Michal Pilipczuk, Jean-Florent Raymond, Dimitrios M. Thilikos, and Marcin Wrochna. Cutwidth: Obstructions and Algorithmic Aspects. In 11th International Symposium on Parameterized and Exact Computation (IPEC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 63, pp. 15:1-15:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{giannopoulou_et_al:LIPIcs.IPEC.2016.15, author = {Giannopoulou, Archontia C. and Pilipczuk, Michal and Raymond, Jean-Florent and Thilikos, Dimitrios M. and Wrochna, Marcin}, title = {{Cutwidth: Obstructions and Algorithmic Aspects}}, booktitle = {11th International Symposium on Parameterized and Exact Computation (IPEC 2016)}, pages = {15:1--15:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-023-1}, ISSN = {1868-8969}, year = {2017}, volume = {63}, editor = {Guo, Jiong and Hermelin, Danny}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2016.15}, URN = {urn:nbn:de:0030-drops-69306}, doi = {10.4230/LIPIcs.IPEC.2016.15}, annote = {Keywords: cutwidth, obstructions, immersions, fixed-parameter tractability} }

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