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

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.