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Abstract: We study a conjecture due to Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, 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 at least 3. Our proof uses a generalization of the concept of brambles to tree-independence number. As a consequence of our main result, several important 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.
Keywords: induced minor, wheel, tree-independence number, Maximum Independent Set
Published in RUP: 25.03.2026; Views: 593; Downloads: 2
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