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arXiv · 2501.00161

Induced Minor Models. II. Sufficient conditions for polynomial-time detection of induced minors

Clément Dallard·Maël Dumas·Claire Hilaire·Anthony Perez
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Abstract

The $H$-Induced Minor Containment problem ($H$-IMC) consists in deciding if a fixed graph $H$ is an induced minor of a graph $G$ given as input, that is, whether $H$ can be obtained from $G$ by deleting vertices and contracting edges. Equivalently, the problem asks if there exists an induced minor model of $H$ in $G$, that is, a collection of disjoint subsets of vertices of $G$, each inducing a connected subgraph, such that contracting each subgraph into a single vertex results in $H$. It is known that $H$-IMC is NP-complete for several graphs $H$, even when $H$ is a tree. In this work, we investigate which properties of $H$ guarantee the existence of an induced minor model whose structure can be leveraged to solve the problem in polynomial time. This allows us to identify four infinite families of graphs $H$ that enjoy such properties. Moreover, we show that if the input graph $G$ excludes long induced paths, then $H$-IMC is polynomial-time solvable for any fixed graph $H$. As a byproduct of our results, this implies that $H$-IMC is polynomial-time solvable for all graphs $H$ with at most $5$ vertices, except for three open cases.

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Clément Dallard, Maël Dumas, Claire Hilaire, Anthony Perez. 2025-10-28. Induced Minor Models. II. Sufficient conditions for polynomial-time detection of induced minors. https://arxiv.org/abs/2501.00161

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