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

TL;DR

This work analyzes the fixed-graph induced minor containment problem $H$-IMC and identifies structural conditions on $H$ or the input graph $G$ that enable polynomial-time detection of an induced minor. It introduces four infinite H-classes (flowers, generalized houses and bulls, complete split graphs $S_{k,p}$ with $k\le 3$, and $K$-NT graphs with a universal clique) for which $H$-IMC is tractable, and proves that $H$-IMC is polynomial on $P_t$-free graphs for any fixed $H$. The authors further derive polynomial-time solvability for Gem-IMC and $\widehat{K_4}$-IMC on general graphs by leveraging $P_t$-free reductions via structural decompositions. Overall, the paper extends the frontier of tractable induced-minor containment problems by coupling minimal-model structure with restricted graph classes, and it provides a unified framework that subsumes several prior results while yielding new polynomial-time algorithms.

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.

Figures (4)

• Figure 1: Exhaustive list of graphs with $5$ vertices. The group of graphs with green background belongs to infinite families studied in this paper. The ones with blue background are the ones for which the complexity of $H$-IMC remains open.
• Figure 2: Note that removing one edge $b_ib_{i+1}$ for some $1\leqslant i \leqslant r-1$ results in a generalized bull.
• Figure 3: (a) Construction of a model for the house with only one non-trivial bag. (b) Example of a graph that admits the bull with subdivided horns as induced minor, but with at least two big bags in every model.
• Figure 4: From left to right: the prism; models of $\widehat{K_4}$ in a subdivided prism and a subdivided $K_{3,3}$; models of $\widehat{K_4}$ for graphs with the \ref{['item:fh4']} in \ref{['thm:fullhouse']}.

Theorems & Definitions (38)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 3.1: $S$-non-trivial property
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4