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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2025.4

Induced Disjoint Paths Without an Induced Minor

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

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

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.

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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)

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@InProceedings{aboulker_et_al:LIPIcs.ICALP.2025.4,
  author =	{Aboulker, Pierre and Bonnet, \'{E}douard and Picavet, Timoth\'{e} and Trotignon, Nicolas},
  title =	{{Induced Disjoint Paths Without an Induced Minor}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{4:1--4:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.4},
  URN =		{urn:nbn:de:0030-drops-233813},
  doi =		{10.4230/LIPIcs.ICALP.2025.4},
  annote =	{Keywords: Induced Disjoint Paths, string graphs, induced subdivisions, induced minors}
}

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DOI: 10.4230/LIPIcs.IPEC.2019.21

The Independent Set Problem Is FPT for Even-Hole-Free Graphs

Authors: Edin Husić, Stéphan Thomassé, and Nicolas Trotignon

Published in: LIPIcs, Volume 148, 14th International Symposium on Parameterized and Exact Computation (IPEC 2019)

Abstract

The class of even-hole-free graphs is very similar to the class of perfect graphs, and was indeed a cornerstone in the tools leading to the proof of the Strong Perfect Graph Theorem. However, the complexity of computing a maximum independent set (MIS) is a long-standing open question in even-hole-free graphs. From the hardness point of view, MIS is W[1]-hard in the class of graphs without induced 4-cycle (when parameterized by the solution size). Halfway of these, we show in this paper that MIS is FPT when parameterized by the solution size in the class of even-hole-free graphs. The main idea is to apply twice the well-known technique of augmenting graphs to extend some initial independent set.

Cite as

Edin Husić, Stéphan Thomassé, and Nicolas Trotignon. The Independent Set Problem Is FPT for Even-Hole-Free Graphs. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 21:1-21:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{husic_et_al:LIPIcs.IPEC.2019.21,
  author =	{Husi\'{c}, Edin and Thomass\'{e}, St\'{e}phan and Trotignon, Nicolas},
  title =	{{The Independent Set Problem Is FPT for Even-Hole-Free Graphs}},
  booktitle =	{14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
  pages =	{21:1--21:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-129-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{148},
  editor =	{Jansen, Bart M. P. and Telle, Jan Arne},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.21},
  URN =		{urn:nbn:de:0030-drops-114826},
  doi =		{10.4230/LIPIcs.IPEC.2019.21},
  annote =	{Keywords: independent set, FPT algorithm, even-hole-free graph, augmenting graph}
}

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Documents authored by Trotignon, Nicolas

Induced Disjoint Paths Without an Induced Minor

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The Independent Set Problem Is FPT for Even-Hole-Free Graphs

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