Excluding an induced wheel minor in graphs without large induced stars

TL;DR

This paper proves a conjecture that certain classes of graphs excluding large induced wheel minors have bounded tree-independence number, leading to tractability of some NP-hard problems and providing algorithms for detecting wheel minors.

Contribution

The paper confirms the conjecture for $k$-wheel minors in $K_{1,d}$-free graphs and introduces a generalized bramble concept to analyze tree-independence number.

Findings

Bounded tree-independence number for graphs excluding large induced wheel minors.

NP-hard problems become tractable on these graph classes.

Polynomial-time algorithms for detecting induced wheel minors.

Abstract

We study a conjecture due to Dallard, Krnc, Kwon, Milani\v{c}, Munaro, \v{S}torgel, and Wiederrecht stating that for any positive integer $d$ and any planar graph $H$, the class of all $K_{1,d}$-free graphs without $H$ as an induced minor has bounded tree-independence number. A $k$-wheel is the graph obtained from a cycle of length $k$ by adding a vertex adjacent to all vertices of the cycle. We show that the conjecture of Dallard et al. is true when $H$ is a $k$-wheel for any $k\geq 3$. Our proof uses a generalization of the concept of brambles to tree-independence number. As a consequence of our main result, several important $\mathsf{NP}$-hard problems such as Maximum Independent Set are tractable on $K_{1,d}$-free graphs without large induced wheel minors. Moreover, for fixed $d$ and $k$, we provide a polynomial-time algorithm that, given a $K_{1,d}$-free graph $G$ as input, finds an induced minor model of a $k$-wheel in $G$ if one exists.