Computing Shrub-Depth Decompositions

Authors Jakub Gajarský, Stephan Kreutzer

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Author Details

Jakub Gajarský
  • Technical University Berlin, Germany
Stephan Kreutzer
  • Technical University Berlin, Germany


We want to thank Sang-il Oum for suggesting that our techniques may be used to obtain the results of Section 6.

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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)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Mathematics of computing → Combinatorial algorithms
  • shrub-depth
  • tree-model
  • decomposition
  • fixed-parameter tractability


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  1. Hans L. Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput., 25(6):1305-1317, 1996. URL:
  2. Bruno Courcelle, Johann A. Makowsky, and Udi Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst., 33(2):125-150, 2000. URL:
  3. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  4. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012. Google Scholar
  5. Jakub Gajarský and Petr Hlinený. Kernelizing MSO properties of trees of fixed height, and some consequences. Logical Methods in Computer Science, 11(1), 2015. URL:
  6. Jakub Gajarský, Stephan Kreutzer, Jaroslav Nesetril, Patrice Ossona de Mendez, Michal Pilipczuk, Sebastian Siebertz, and Szymon Toru'nczyk. First-order interpretations of bounded expansion classes. In Ioannis Chatzigiannakis, Christos Kaklamanis, Dániel Marx, and Donald Sannella, editors, 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, volume 107 of LIPIcs, pages 126:1-126:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. URL:
  7. Jakub Gajarský, Michael Lampis, and Sebastian Ordyniak. Parameterized algorithms for modular-width. In Gregory Z. Gutin and Stefan Szeider, editors, Parameterized and Exact Computation - 8th International Symposium, IPEC 2013, Sophia Antipolis, France, September 4-6, 2013, Revised Selected Papers, volume 8246 of Lecture Notes in Computer Science, pages 163-176. Springer, 2013. URL:
  8. Robert Ganian, Petr Hlinený, Jaroslav Nešetřil, Jan Obdržálek, and Patrice Ossona de Mendez. Shrub-depth: Capturing height of dense graphs. Logical Methods in Computer Science, 15(1), 2019. URL:, URL:
  9. Petr Hlinený and Sang-il Oum. Finding branch-decompositions and rank-decompositions. SIAM J. Comput., 38(3):1012-1032, 2008. URL:
  10. Jaroslav Nesetril and Patrice Ossona de Mendez. Tree-depth, subgraph coloring and homomorphism bounds. Eur. J. Comb., 27(6):1022-1041, 2006. URL:
  11. Jaroslav Nesetril and Patrice Ossona de Mendez. Sparsity - Graphs, Structures, and Algorithms, volume 28 of Algorithms and combinatorics. Springer, 2012. URL:
  12. Felix Reidl, Peter Rossmanith, Fernando Sánchez Villaamil, and Somnath Sikdar. A faster parameterized algorithm for treedepth. In Javier Esparza, Pierre Fraigniaud, Thore Husfeldt, and Elias Koutsoupias, editors, Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part I, volume 8572 of Lecture Notes in Computer Science, pages 931-942. Springer, 2014. URL:
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