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Induced Disjoint Paths Without an Induced Minor

Authors Pierre Aboulker , Édouard Bonnet , Timothé Picavet , Nicolas Trotignon

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ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Mathematics of computing → Graph theory
Keywords
  • Induced Disjoint Paths
  • string graphs
  • induced subdivisions
  • induced minors

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Abstract

We exhibit a new obstacle to the nascent algorithmic theory for classes excluding an induced minor. We indeed show that on the class of string graphs - which avoids the 1-subdivision of, say, K₅ as an induced minor - Induced 2-Disjoint Paths is NP-complete. So, while k-Disjoint Paths, for a fixed k, is polynomial-time solvable in general graphs, the absence of a graph as an induced minor does not make its induced variant tractable, even for k = 2. This answers a question of Korhonen and Lokshtanov [SODA '24], and complements a polynomial-time algorithm for Induced k-Disjoint Paths in classes of bounded genus by Kobayashi and Kawarabayashi [SODA '09]. In addition to being string graphs, our produced hard instances are subgraphs of a constant power of bounded-degree planar graphs, hence have bounded twin-width and bounded maximum degree.
We also leverage our new result to show that there is a fixed subcubic graph H such that deciding if an input graph contains H as an induced subdivision is NP-complete. Until now, all the graphs H for which such a statement was known had a vertex of degree at least 4. This answers a question by Chudnovsky, Seymour, and Trotignon [JCTB '13], and by Le [JGT '19]. Finally we resolve another question of Korhonen and Lokshtanov by exhibiting a subcubic graph H without two adjacent degree-3 vertices and such that deciding if an input n-vertex graph contains H as an induced minor is NP-complete, and unless the Exponential-Time Hypothesis fails, requires time 2^{Ω(√ n)}. This complements an algorithm running in subexponential time 2^{Õ(n^{2/3})} by these authors [SODA '24] under the same technical condition.

Cite As Get BibTex

Pierre Aboulker, Édouard Bonnet, Timothé Picavet, and Nicolas Trotignon. Induced Disjoint Paths Without an Induced Minor. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 4:1-4:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.4

Author Details

Pierre Aboulker
  • DIENS, École normale supérieure, CNRS, PSL University, Paris, France
Édouard Bonnet
  • CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR 5668, Lyon, France
Timothé Picavet
  • LaBRI, Université de Bordeaux, France
Nicolas Trotignon
  • CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR 5668, Lyon, France

Funding

ÉB and NT were supported by the ANR projects TWIN-WIDTH (ANR-21-CE48-0014-01) and DIGRAPHS (ANR-19-CE48-0013-01).

Acknowledgements

We thank Théo Pierron and Jean-Florent Raymond with whom this project started. We also thank Tuukka Korhonen, Daniel Lokshtanov, and Martin Milanič for some discussions, and in particular for asking about the flow problem (as opposed to the linkage one).

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