Induced Disjoint Paths Without an Induced Minor
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2025
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| author | Aboulker, Pierre Bonnet, Édouard Picavet, Timothé Trotignon, Nicolas |
| author_facet | Aboulker, Pierre Bonnet, Édouard Picavet, Timothé Trotignon, Nicolas |
| contents | 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, $K_5$ 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 the fourth author [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 $2^{Ω(\sqrt n)}$. This complements an algorithm running in subexponential time $2^{O(n^{2/3} \log n)}$ by these authors [SODA '24] under the same technical condition. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_05289 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Induced Disjoint Paths Without an Induced Minor Aboulker, Pierre Bonnet, Édouard Picavet, Timothé Trotignon, Nicolas Computational Complexity Discrete Mathematics Combinatorics 68Q25 F.2.2 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, $K_5$ 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 the fourth author [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 $2^{Ω(\sqrt n)}$. This complements an algorithm running in subexponential time $2^{O(n^{2/3} \log n)}$ by these authors [SODA '24] under the same technical condition. |
| title | Induced Disjoint Paths Without an Induced Minor |
| topic | Computational Complexity Discrete Mathematics Combinatorics 68Q25 F.2.2 |
| url | https://arxiv.org/abs/2502.05289 |