Skip to main content

Few Induced Disjoint Paths for H-Free Graphs

  • Conference paper
  • First Online:
Combinatorial Optimization (ISCO 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13526))

Included in the following conference series:

Abstract

Paths \(P^1,\ldots ,P^k\) in a graph \(G=(V,E)\) are mutually induced if any two distinct \(P^i\) and \(P^j\) have neither common vertices nor adjacent vertices. For a fixed integer k, the k-Induced Disjoint Paths problem is to decide if a graph G with k pairs of specified vertices \((s_i,t_i)\) contains k mutually induced paths \(P^i\) such that each \(P^i\) starts from \(s_i\) and ends at \(t_i\). Whereas the non-induced version is well-known to be polynomial-time solvable for every fixed integer k, a classical result from the literature states that even 2-Induced Disjoint Paths is NP-complete. We prove new complexity results for k-Induced Disjoint Paths if the input is restricted to H-free graphs, that is, graphs without a fixed graph H as an induced subgraph. We compare our results with a complexity dichotomy for Induced Disjoint Paths, the variant where k is part of the input.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    See https://github.com/barnabymartin/InducedSubgraph.

References

  1. Alekseev, V.E.: Polynomial algorithm for finding the largest independent sets in graphs without forks. Discrete Appl. Math. 135, 3–16 (2004)

    Article  MathSciNet  Google Scholar 

  2. Belmonte, R., Golovach, P.A., Heggernes, P., van’t Hof, P., Kaminski, M., Paulusma, D.: Detecting fixed patterns in chordal graphs in polynomial time. Algorithmica 69, 501–521 (2014). https://doi.org/10.1007/s00453-013-9748-5

  3. Bienstock, D.: On the complexity of testing for odd holes and induced odd paths. Discrete Math. 90, 85–92 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Fellows, M.R.: The Robertson-Seymour theorems: a survey of applications. In: Proceedings of AMS-IMS-SIAM Joint Summer Research Conference (1989). Contemp. Math. 89, 1–18

    Google Scholar 

  5. Fiala, J., Kamiński, M., Lidický, B., Paulusma, D.: The \(k\)-in-a-path problem for claw-free graphs. Algorithmica 62, 499–519 (2012). https://doi.org/10.1007/s00453-010-9468-z

    Article  MathSciNet  MATH  Google Scholar 

  6. Golovach, P.A., Paulusma, D., van Leeuwen, E.J.: Induced disjoint paths in claw-free graphs. SIAM J. Discrete Math. 29, 348–375 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Golovach, P.A., Paulusma, D., van Leeuwen, E.J.: Induced disjoint paths in circular-arc graphs in linear time. Theor. Comput. Sci. 640, 70–83 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Golovach, P.A., Paulusma, D., van Leeuwen, E.J.: Induced disjoint paths in AT-free graphs. J. Comput. Syst. Sci. 124, 170–191 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jaffke, L., Kwon, O., Telle, J.A.: Mim-width I. induced path problems. Discrete Appl. Math. 278, 153–168 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kawarabayashi, K., Kobayashi, Y.: A linear time algorithm for the induced disjoint paths problem in planar graphs. J. Comput. Syst. Sci. 78, 670–680 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kern, W., Martin, B., Paulusma, D., Smith, S., van Leeuwen, E.J.: Disjoint paths and connected subgraphs for \(H\)-free graphs. Theor. Comput. Sci. 898, 59–68 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kobayashi, Y., Kawarabayashi, K.: Algorithms for finding an induced cycle in planar graphs and bounded genus graphs. In: Proceedings of the SODA 2009, pp. 1146–1155 (2009)

    Google Scholar 

  13. Lévêque, B., Lin, D.Y., Maffray, F., Trotignon, N.: Detecting induced subgraphs. Discrete Appl. Math. 157, 3540–3551 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lynch, J.: The equivalence of theorem proving and the interconnection problem. SIGDA Newslett. 5, 31–36 (1975)

    Article  Google Scholar 

  15. Martin, B., Paulusma, D., Smith, S., van Leeuwen, E.J.: Induced disjoint paths and connected subgraphs for \({H}\)-free graphs. In: Bekos, M.A., Kaufmann, M. (eds.) WG 2022. LNCS, vol. 13453, pp. 398–411. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-15914-5_29

    Chapter  Google Scholar 

  16. Radovanović, M., Trotignon, N., Vus̆ković, K.: The (theta, wheel)-free graphs Part IV: induced paths and cycles. J. Comb. Theory Ser. B 146, 495–531 (2021)

    Google Scholar 

  17. Robertson, N., Seymour, P.D.: Graph minors. XIII. The disjoint paths problem. J. Comb. Theory Ser. B 63, 65–110 (1995)

    Google Scholar 

  18. Shibi, N.: Algorithme de recherche d’un stable de cardinalité maximum dans un graphe sans étoile. Discrete Math. 29, 53–76 (1980)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Barnaby Martin .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Martin, B., Paulusma, D., Smith, S., van Leeuwen, E.J. (2022). Few Induced Disjoint Paths for H-Free Graphs. In: Ljubić, I., Barahona, F., Dey, S.S., Mahjoub, A.R. (eds) Combinatorial Optimization. ISCO 2022. Lecture Notes in Computer Science, vol 13526. Springer, Cham. https://doi.org/10.1007/978-3-031-18530-4_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-18530-4_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-18529-8

  • Online ISBN: 978-3-031-18530-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics