Bridge-Depth Characterizes Which Structural Parameterizations of Vertex Cover Admit a Polynomial Kernel

Authors Marin Bougeret , Bart M. P. Jansen , Ignasi Sau

Thumbnail PDF


  • Filesize: 1.23 MB
  • 19 pages

Document Identifiers

Author Details

Marin Bougeret
  • LIRMM, Université de Montpellier, CNRS, France
Bart M. P. Jansen
  • Eindhoven University of Technology, The Netherlands
Ignasi Sau
  • LIRMM, Université de Montpellier, CNRS, France

Cite AsGet BibTex

Marin Bougeret, Bart M. P. Jansen, and Ignasi Sau. Bridge-Depth Characterizes Which Structural Parameterizations of Vertex Cover Admit a Polynomial Kernel. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 16:1-16:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We study the kernelization complexity of structural parameterizations of the Vertex Cover problem. Here, the goal is to find a polynomial-time preprocessing algorithm that can reduce any instance (G,k) of the Vertex Cover problem to an equivalent one, whose size is polynomial in the size of a pre-determined complexity parameter of G. A long line of previous research deals with parameterizations based on the number of vertex deletions needed to reduce G to a member of a simple graph class ℱ, such as forests, graphs of bounded tree-depth, and graphs of maximum degree two. We set out to find the most general graph classes ℱ for which Vertex Cover parameterized by the vertex-deletion distance of the input graph to ℱ, admits a polynomial kernelization. We give a complete characterization of the minor-closed graph families ℱ for which such a kernelization exists. We introduce a new graph parameter called bridge-depth, and prove that a polynomial kernelization exists if and only if ℱ has bounded bridge-depth. The proof is based on an interesting connection between bridge-depth and the size of minimal blocking sets in graphs, which are vertex sets whose removal decreases the independence number.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Parameterized complexity and exact algorithms
  • vertex cover
  • parameterized complexity
  • polynomial kernel
  • structural parameterization
  • bridge-depth


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  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. Hans L. Bodlaender. A partial k-arboretum of graphs with bounded treewidth. Theor. Comput. Sci., 209(1-2):1-45, 1998. URL:
  3. Hans L. Bodlaender. Kernelization, exponential lower bounds. In Encyclopedia of Algorithms, pages 1013-1017. Springer, 2016. URL:
  4. Marin Bougeret, Bart M. P. Jansen, and Ignasi Sau. Bridge-Depth Characterizes which Structural Parameterizations of Vertex Cover Admit a Polynomial Kernel. CoRR, abs/2004.12865, 2020. URL:
  5. Marin Bougeret and Ignasi Sau. How much does a treedepth modulator help to obtain polynomial kernels beyond sparse graphs? Algorithmica, 81(10):4043-4068, 2019. URL:
  6. Jannis Bulian and Anuj Dawar. Graph isomorphism parameterized by elimination distance to bounded degree. Algorithmica, 75(2):363-382, 2016. URL:
  7. Li-Hsuan Chen, Felix Reidl, Peter Rossmanith, and Fernando Sánchez Villaamil. Width, depth, and space: Tradeoffs between branching and dynamic programming. Algorithms, 11(7):98, 2018. URL:
  8. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL:
  9. Marek Cygan, Daniel Lokshtanov, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. On the hardness of losing width. Theory Comput. Syst., 54(1):73-82, 2014. URL:
  10. Holger Dell and Dieter van Melkebeek. Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. J. ACM, 61(4):23:1-23:27, 2014. URL:
  11. Reinhard Diestel. Graph Theory. Springer-Verlag, Heidelberg, 5th edition, 2016. Google Scholar
  12. Rod G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. Google Scholar
  13. Michael R. Fellows, Lars Jaffke, Aliz Izabella Király, Frances A. Rosamond, and Mathias Weller. What is known about vertex cover kernelization? In Hans-Joachim Böckenhauer, Dennis Komm, and Walter Unger, editors, Adventures Between Lower Bounds and Higher Altitudes - Essays Dedicated to Juraj Hromkovič on the Occasion of His 60th Birthday, volume 11011 of Lecture Notes in Computer Science, pages 330-356. Springer, 2018. URL:
  14. Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Springer-Verlag, 2006. URL:
  15. Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, Geevarghese Philip, and Saket Saurabh. Hitting forbidden minors: Approximation and kernelization. SIAM J. Discrete Math., 30(1):383-410, 2016. URL:
  16. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Kernelization: Theory of Parameterized Preprocessing. Cambridge University Press, 2019. URL:
  17. Fedor V. Fomin and Torstein J. F. Strømme. Vertex cover structural parameterization revisited. In Proc. 42nd WG, volume 9941 of LNCS, pages 171-182, 2016. URL:
  18. Eva-Maria C. Hols and Stefan Kratsch. Smaller parameters for vertex cover kernelization. In Proc. 12th IPEC, volume 89 of LIPIcs, pages 20:1-20:12, 2017. URL:
  19. Eva-Maria C. Hols, Stefan Kratsch, and Astrid Pieterse. Elimination distances, blocking sets, and kernels for vertex cover. In Proc. 37nd STACS, volume 154 of LIPIcs, pages 36:1-36:14, 2020. URL:
  20. John E. Hopcroft and Robert Endre Tarjan. Efficient algorithms for graph manipulation (algorithm 447). Commun. ACM, 16(6):372-378, 1973. URL:
  21. Bart M. P. Jansen and Hans L. Bodlaender. Vertex cover kernelization revisited: Upper and lower bounds for a refined parameter. In Proc. 28th STACS, volume 9 of LIPIcs, pages 177-188, 2011. URL:
  22. Bart M. P. Jansen and Hans L. Bodlaender. Vertex cover kernelization revisited - upper and lower bounds for a refined parameter. Theory Comput. Syst., 53(2):263-299, 2013. URL:
  23. Bart M. P. Jansen and Astrid Pieterse. Polynomial kernels for hitting forbidden minors under structural parameterizations. In Proc. 26th ESA, volume 112 of LIPIcs, pages 48:1-48:15, 2018. URL:
  24. Stefan Kratsch. A randomized polynomial kernelization for vertex cover with a smaller parameter. SIAM J. Discrete Math., 32(3):1806-1839, 2018. URL:
  25. Stefan Kratsch and Magnus Wahlström. Representative sets and irrelevant vertices: New tools for kernelization. In Proc. 53rd FOCS, pages 450-459, 2012. URL:
  26. Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. Kernelization - Preprocessing with a guarantee. In The Multivariate Algorithmic Revolution and Beyond, pages 129-161, 2012. URL:
  27. Diptapriyo Majumdar, Venkatesh Raman, and Saket Saurabh. Polynomial kernels for vertex cover parameterized by small degree modulators. Theory Comput. Syst., 62(8):1910-1951, 2018. URL:
  28. George L. Nemhauser and Leslie E. Trotter Jr. Vertex packings: structural properties and algorithms. Math. Program., 8:232-248, 1975. URL:
  29. Jaroslav Nesetril and Patrice Ossona de Mendez. Sparsity - Graphs, Structures, and Algorithms, volume 28 of Algorithms and combinatorics. Springer, 2012. URL:
  30. Jaroslav Nesetril and Patrice Ossona de Mendez. Tree-depth, subgraph coloring and homomorphism bounds. Eur. J. Comb., 27(6):1022-1041, 2006. URL:
  31. Rolf Niedermeier. Invitation to Fixed-Parameter Algorithms. Oxford University Press, 2006. URL:
  32. Michal Pilipczuk and Marcin Wrochna. On space efficiency of algorithms working on structural decompositions of graphs. ACM TOCT, 9(4):18:1-18:36, 2018. URL:
  33. Dieter Rautenbach and Bruce A. Reed. The Erdös-Pósa Property for Odd Cycles in Highly Connected Graphs. Combinatorica, 21(2):267-278, 2001. URL:
  34. Neil Robertson and Paul D. Seymour. Graph minors. V. Excluding a planar graph. J. Comb. Theory, Ser. B, 41(1):92-114, 1986. URL:
  35. Neil Robertson and Paul D. Seymour. Graph minors. XX. Wagner’s conjecture. J. Comb. Theory, Ser. B, 92(2):325-357, 2004. URL:
  36. Wouter Cames van Batenburg, Tony Huynh, Gwenaël Joret, and Jean-Florent Raymond. A tight Erdös-Pósa function for planar minors. In Proc. 30th SODA, pages 1485-1500. SIAM, 2019. URL: