Document Open Access Logo

On Graphs Coverable by k Shortest Paths

Authors Maël Dumas, Florent Foucaud , Anthony Perez, Ioan Todinca

PDF
Thumbnail PDF

File

LIPIcs.ISAAC.2022.40.pdf
  • Filesize: 0.92 MB
  • 15 pages

Document Identifiers

Author Details

Maël Dumas
  • Univ. Orléans, INSA Centre Val de Loire, LIFO EA 4022, F-45067 Orléans, France
Florent Foucaud
  • Université Clermont-Auvergne, CNRS, Mines de Saint-Étienne, Clermont-Auvergne-INP, LIMOS, 63000 Clermont-Ferrand, France
Anthony Perez
  • Univ. Orléans, INSA Centre Val de Loire, LIFO EA 4022, F-45067 Orléans, France
Ioan Todinca
  • Univ. Orléans, INSA Centre Val de Loire, LIFO EA 4022, F-45067 Orléans, France

Funding

  • Foucaud, Florent: This author was financed by the IFCAM project "Applications of graph homomorphisms" (MA/IFCAM/18/39), the ANR project GRALMECO (ANR-21-CE48-0004-01) and the French government IDEX-ISITE initiative 16-IDEX-0001 (CAP 20-25).

Cite AsGet BibTex

Maël Dumas, Florent Foucaud, Anthony Perez, and Ioan Todinca. On Graphs Coverable by k Shortest Paths. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 40:1-40:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ISAAC.2022.40

Abstract

We show that if the edges or vertices of an undirected graph G can be covered by k shortest paths, then the pathwidth of G is upper-bounded by a function of k. As a corollary, we prove that the problem Isometric Path Cover with Terminals (which, given a graph G and a set of k pairs of vertices called terminals, asks whether G can be covered by k shortest paths, each joining a pair of terminals) is FPT with respect to the number of terminals. The same holds for the similar problem Strong Geodetic Set with Terminals (which, given a graph G and a set of k terminals, asks whether there exist binom(k,2) shortest paths, each joining a distinct pair of terminals such that these paths cover G). Moreover, this implies that the related problems Isometric Path Cover and Strong Geodetic Set (defined similarly but where the set of terminals is not part of the input) are in XP with respect to parameter k.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Shortest paths
  • covering problems
  • parameterized complexity

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. M. Aigner and M. Fromme. A game of cops and robbers. Discrete Applied Mathematics, 8(1):1-12, 1984. URL: https://doi.org/10.1016/0166-218X(84)90073-8.
  2. Giovanni Andreatta and Francesco Mason. Path covering problems and testing of printed circuits. Discrete Applied Mathematics, 62(1):5-13, 1995. URL: https://doi.org/10.1016/0166-218X(94)00142-Z.
  3. Stefan Arnborg, Jens Lagergren, and Detlef Seese. Easy problems for tree-decomposable graphs. Journal of Algorithms, 12(2):308-340, 1991. URL: https://doi.org/10.1016/0196-6774(91)90006-K.
  4. Andrew Arokiaraj, Sandi Klavžar, Paul D. Manuel, Elizabeth Thomas, and Antony Xavier. Strong geodetic problems in networks. Discussiones Mathematicae Graph Theory, 40(1):307-321, 2020. URL: https://doi.org/10.7151/dmgt.2139.
  5. Dibyayan Chakraborty, Antoine Dailly, Sandip Das, Florent Foucaud, Harmender Gahlawat, and Subir K. Ghosh. Complexity and algorithms for isometric path cover on chordal graphs and beyond. In Proceedings of the 33rd International Symposium on Algorithms and Computation, ISAAC 2022, LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. Google Scholar
  6. Bruno Courcelle. The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Information and Computation, 85(1):12-75, 1990. URL: https://doi.org/10.1016/0890-5401(90)90043-H.
  7. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  8. Tom Davot, Lucas Isenmann, and Jocelyn Thiebaut. On the approximation hardness of geodetic set and its variants. In Proceedings of the 27th International Computing and Combinatorics Conference, COCOON 2021, volume 13025 of Lecture Notes in Computer Science, pages 76-88. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-89543-3_7.
  9. João Henrique Gonçalves de Sousa. Exact algorithms and computational complexity for the strong geodetic set problem. Master’s thesis, Universidade Federal de Minas Gerais, Brazil, 2018. URL: http://hdl.handle.net/1843/ESBF-BA8NJD.
  10. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012. Google Scholar
  11. Tali Eilam-Tzoreff. The disjoint shortest paths problem. Discrete Applied Mathematics, 85(2):113-138, 1998. URL: https://doi.org/10.1016/S0166-218X(97)00121-2.
  12. David C. Fisher and Shannon L. Fitzpatrick. The isometric number of a graph. Journal of Combinatorial Mathematics and Combinatorial Computing, 38(1):97-110, 2001. Google Scholar
  13. Ioannis Katsikarelis, Michael Lampis, and Vangelis Th. Paschos. Structurally parameterized d-scattered set. Discret. Appl. Math., 308:168-186, 2022. URL: https://doi.org/10.1016/j.dam.2020.03.052.
  14. Leon Kellerhals and Tomohiro Koana. Parameterized complexity of geodetic set. In Proceedings of the 15th International Symposium on Parameterized and Exact Computation, IPEC 2020, volume 180 of LIPIcs, pages 20:1-20:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.IPEC.2020.20.
  15. Carlos V. G. C. Lima, Vinicius F. dos Santos, João H. G. Sousa, and Sebastián A. Urrutia. On the computational complexity of the strong geodetic recognition problem, 2022. URL: https://doi.org/10.48550/ARXIV.2208.01796.
  16. Willian Lochet. A polynomial time algorithm for the k-disjoint shortest paths problem. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, pages 169-178. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.12.
  17. Paul Manuel. On the isometric path partition problem. Discussiones Mathematicae Graph Theory, 41(4):1077-1089, 2021. URL: https://doi.org/10.7151/dmgt.2236.
  18. Paul Manuel, Sandi Klavžar, Antony Xavier, Andrew Arokiaraj, and Elizabeth Thomas. Strong edge geodetic problem in networks. Open Mathematics, 15(1):1225-1235, 2017. URL: https://doi.org/10.1515/math-2017-0101.
  19. Mauro Mezzini. An O(mn²) algorithm for computing the strong geodetic number in outerplanar graphs. Discussiones Mathematicae Graph Theory, 42(2):591-599, 2022. URL: https://doi.org/10.7151/dmgt.2311.
  20. Jun-Jie Pan and Gerard J. Chang. Isometric-path numbers of block graphs. Information Processing Letters, 93(2):99-102, 2005. URL: https://doi.org/10.1016/j.ipl.2004.09.021.
  21. Neil Robertson and Paul D. Seymour. Graph minors. XIII. The disjoint paths problem. Journal of Combinatorial Theory, Series B, 63(1):65-110, 1995. URL: https://doi.org/10.1006/jctb.1995.1006.
  22. Maximilian Thiessen and Thomas Gaertner. Active learning of convex halfspaces on graphs. In Proceedings of the 35th Conference on Neural Information Processing Systems, NeurIPS 2021, volume 34, pages 23413-23425. Curran Associates, Inc., 2021. URL: https://proceedings.neurips.cc/paper/2021/file/c4bf1e24f3e6f92ca9dfd9a7a1a1049c-Paper.pdf.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact