Induced Disjoint Paths Without an Induced Minor

2025-02-07

AI Summary

This paper investigates the computational complexity of the Induced Disjoint Paths problem on graph classes excluding induced subgraphs. For Induced 2-Disjoint Paths, we establish its NP-completeness on chordal graphs, graphs of bounded twin-width, and graphs of bounded degree—demonstrating that excluding induced subgraphs alone does not ensure tractability. We further construct the first subcubic graph $ H $ for which both induced subdivision and induced subgraph containment are NP-complete; under the Exponential Time Hypothesis (ETH), this yields a tight $ 2^{Omega(sqrt{n})} $ lower bound, breaking prior restrictions on vertex degree. Our approach integrates combinatorial graph constructions, reductions between induced subgraphs and induced subdivisions, structural analysis of chordal graphs, and refined lower-bound arguments. These techniques resolve several long-standing open problems in parameterized and structural graph algorithms.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Bridging structured and unstructured data

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, $K_5$ 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 $2^{Omega(sqrt n)}$. 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.

Problem

• NP-completeness of Induced 2-Disjoint Paths in string graphs
• Fixed subcubic graph induced subdivision is NP-complete
• NP-completeness of induced minor in subcubic graphs

Innovation

• NP-complete in string graphs
• Bounded twin-width instances
• Fixed subcubic graph NP-complete