Document Open Access Logo

Parameterized Complexity of Non-Separating and Non-Disconnecting Paths and Sets

Authors Ankit Abhinav, Susobhan Bandopadhyay, Aritra Banik, Yasuaki Kobayashi , Shunsuke Nagano, Yota Otachi , Saket Saurabh



PDF
Thumbnail PDF

File

LIPIcs.MFCS.2022.6.pdf
  • Filesize: 0.87 MB
  • 15 pages

Document Identifiers

Author Details

Ankit Abhinav
  • National Institute of Science, Education and Research, An OCC of Homi Bhabha National Institute, Bhubaneswar, Odisha, India
Susobhan Bandopadhyay
  • National Institute of Science, Education and Research, An OCC of Homi Bhabha National Institute, Bhubaneswar, Odisha, India
Aritra Banik
  • National Institute of Science, Education and Research, An OCC of Homi Bhabha National Institute, Bhubaneswar, Odisha, India
Yasuaki Kobayashi
  • Kyoto University, Kyoto, Japan
Shunsuke Nagano
  • Kyoto University, Kyoto, Japan
Yota Otachi
  • Nagoya University, Nagoya, Japan
Saket Saurabh
  • The Institute of Mathematical Sciences, HBNI, Chennai, India

Cite AsGet BibTex

Ankit Abhinav, Susobhan Bandopadhyay, Aritra Banik, Yasuaki Kobayashi, Shunsuke Nagano, Yota Otachi, and Saket Saurabh. Parameterized Complexity of Non-Separating and Non-Disconnecting Paths and Sets. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 6:1-6:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.MFCS.2022.6

Abstract

For a connected graph G = (V, E) and s, t ∈ V, a non-separating s-t path is a path P between s and t such that the set of vertices of P does not separate G, that is, G - V(P) is connected. An s-t path P is non-disconnecting if G - E(P) is connected. The problems of finding shortest non-separating and non-disconnecting paths are both known to be NP-hard. In this paper, we consider the problems from the viewpoint of parameterized complexity. We show that the problem of finding a non-separating s-t path of length at most k is W[1]-hard parameterized by k, while the non-disconnecting counterpart is fixed-parameter tractable (FPT) parameterized by k. We also consider the shortest non-separating path problem on several classes of graphs and show that this problem is NP-hard even on bipartite graphs, split graphs, and planar graphs. As for positive results, the shortest non-separating path problem is FPT parameterized by k on planar graphs and on unit disk graphs (where no s, t is given). Further, we give a polynomial-time algorithm on chordal graphs if k is the distance of the shortest path between s and t.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
Keywords
  • Non-separating path
  • Parameterized complexity

Metrics

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

References

  1. Stefan Arnborg, Jens Lagergren, and Detlef Seese. Easy problems for tree-decomposable graphs. J. Algorithms, 12(2):308-340, 1991. URL: https://doi.org/10.1016/0196-6774(91)90006-K.
  2. Hans L. Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput., 25(6):1305-1317, 1996. URL: https://doi.org/10.1137/S0097539793251219.
  3. Hans L. Bodlaender, Rodney G. Downey, Michael R. Fellows, and Danny Hermelin. On problems without polynomial kernels. J. Comput. Syst. Sci., 75(8):423-434, 2009. URL: https://doi.org/10.1016/j.jcss.2009.04.001.
  4. Guantao Chen, Ronald J. Gould, and Xingxing Yu. Graph connectivity after path removal. Comb., 23(2):185-203, 2003. URL: https://doi.org/10.1007/s003-0018-z.
  5. Bruno Courcelle. The monadic second-order logic of graphs. I. recognizable sets of finite graphs. Inf. Comput., 85(1):12-75, 1990. Google Scholar
  6. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Daniel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer Publishing Company, Incorporated, 1st edition, 2015. Google Scholar
  7. Erik D. Demaine and Mohammad Taghi Hajiaghayi. Diameter and treewidth in minor-closed graph families, revisited. Algorithmica, 40(3):211-215, 2004. URL: https://doi.org/10.1007/s00453-004-1106-1.
  8. G. A. Dirac. On rigid circuit graphs. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 25:71-76, 1961. Google Scholar
  9. David Eppstein. Diameter and treewidth in minor-closed graph families. Algorithmica, 27(3):275-291, 2000. URL: https://doi.org/10.1007/s004530010020.
  10. Michael R. Fellows, Danny Hermelin, Frances A. Rosamond, and Stéphane Vialette. On the parameterized complexity of multiple-interval graph problems. Theor. Comput. Sci., 410(1):53-61, 2009. URL: https://doi.org/10.1016/j.tcs.2008.09.065.
  11. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. Efficient computation of representative families with applications in parameterized and exact algorithms. J. ACM, 63(4):29:1-29:60, 2016. Google Scholar
  12. Martin Grohe and Stephan Kreutzer. Methods for algorithmic meta theorems. In AMS-ASL Joint Special Session, volume 558 of Contemporary Mathematics, pages 181-206. American Mathematical Society, 2009. Google Scholar
  13. Martin Grohe, Stephan Kreutzer, and Sebastian Siebertz. Deciding first-order properties of nowhere dense graphs. J. ACM, 64(3):17:1-17:32, 2017. URL: https://doi.org/10.1145/3051095.
  14. Alon Itai, Christos H. Papadimitriou, and Jayme Luiz Szwarcfiter. Hamilton paths in grid graphs. SIAM J. Comput., 11(4):676-686, 1982. URL: https://doi.org/10.1137/0211056.
  15. Ken-ichi Kawarabayashi, Orlando Lee, Bruce A. Reed, and Paul Wollan. A weaker version of lovász' path removal conjecture. J. Comb. Theory, Ser. B, 98(5):972-979, 2008. URL: https://doi.org/10.1016/j.jctb.2007.11.003.
  16. Tuukka Korhonen. A single-exponential time 2-approximation algorithm for treewidth. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 184-192. IEEE, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00026.
  17. Matthias Kriesell. Induced paths in 5-connected graphs. J. Graph Theory, 36(1):52-58, 2001. URL: https://doi.org/10.1002/1097-0118(200101)36:1<52::AID-JGT5>3.0.CO;2-N.
  18. Daniel Lokshtanov, Pranabendu Misra, Fahad Panolan, and Saket Saurabh. Deterministic truncation of linear matroids. ACM Trans. Algorithms, 14(2):14:1-14:20, 2018. Google Scholar
  19. Xiao Mao. Shortest non-separating st-path on chordal graphs. CoRR, abs/2101.03519, 2021. URL: http://arxiv.org/abs/2101.03519.
  20. Haiko Müller. Hamiltonian circuits in chordal bipartite graphs. Discret. Math., 156(1-3):291-298, 1996. Google Scholar
  21. James G. Oxley. Matroid Theory (Oxford Graduate Texts in Mathematics). Oxford University Press, Inc., USA, 2006. Google Scholar
  22. Neil Robertson and Paul D. Seymour. Graph minors. III. planar tree-width. J. Comb. Theory, Ser. B, 36(1):49-64, 1984. URL: https://doi.org/10.1016/0095-8956(84)90013-3.
  23. William T. Tutte. How to draw a graph. Proceedings of the London Mathematical Society, s3-13(1):743-767, 1963. URL: https://doi.org/10.1112/plms/s3-13.1.743.
  24. Bang Ye Wu and Hung-Chou Chen. The approximability of the minimum border problem. In The 26th Workshop on Combinatorial Mathematics and Computation Theory, 2009. Google Scholar
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