Induced Disjoint Paths Without an Induced Minor

TL;DR

We address the tractability frontier for Induced $k$-Disjoint Paths under induced-minor exclusions by proving that Induced $2$-Disjoint Paths is NP-complete on string graphs that are subgraphs of a constant power of bounded-degree planar graphs, hence these exclusions do not suffice for tractability. Our reductions place the problem within a bounded twin-width regime and yield ETH-based lower bounds for related induced-subdivision and induced-minor problems, complementing existing polynomial-time results for bounded-genus classes. We further show that detecting a fixed subcubic graph as an induced subdivision is NP-complete, answering longstanding questions, and extend this to NP-hardness and subexponential lower bounds for H-Induced Minor Containment with a carefully crafted H', while keeping the reductions linear. Together, these results sharpen the understanding of complexity borders for induced-graph containment problems and highlight intrinsic hardness even in structurally restricted graph classes.

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 $2^{Ω(\sqrt n)}$. 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.

Figures (2)

• Figure 1: The subcubic graph $H$. Both $H[A_1]$ and $H[A_2]$ are the 1-subdivision of $K_{3,3}$. Both $H[A_3]$ and $H[A_4]$ are obtained from $K_{3,3}$ by subdividing every edge once but one edge that is subdivided twice. The vertices of $H$ which are not in $\bigcup_{i \in [4]} A_i$ are labeled $s, t, s', t'$. In total $H$ has 66 vertices.
• Figure 5: The graph $H'$ obtained from $H$ of \ref{['fig:H']} by subdividing in each $A_i$ the two edges incident to the vertex with a neighbor in $\{s,t,s',t'\}$. In total $H'$ has 74 vertices.

Theorems & Definitions (8)

• Theorem 1
• Corollary 2
• Theorem 3
• Theorem 4
• Conjecture 5
• Theorem 6: Theorem 6.3 in twin-width1, HlinenyJ23
• Theorem 7: Theorem 8.1 in twin-width1
• Theorem 8: twin-width2