Excluding an induced wheel minor in graphs without large induced stars

2025-06-10

AI Summary

This paper addresses a conjecture by Dallard et al. concerning the boundedness of tree-independence number: for any positive integer $d$ and planar graph $H$, is the tree-independence number bounded for the class of graphs excluding $K_{1,d}$ as an induced subgraph and $H$ as an induced minor? The focus is specifically on $H$ being a $k$-wheel ($k geq 3$). Method: We generalize the notion of bramble to the tree-independence framework and establish a fundamental connection between excluding $k$-wheels as induced minors and structural boundedness. Our approach integrates generalized bramble theory, structural analysis of graph minors, and tree decomposition characterizations. Contribution/Results: We prove that graphs excluding $K_{1,d}$ and $k$-wheels as induced minors have bounded tree-independence number. Consequently, several NP-hard problems—including Maximum Independent Set—become polynomial-time solvable on this graph class. Furthermore, we design a polynomial-time algorithm to detect $k$-wheel induced minors.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study a conjecture due to Dallard, Krnc, Kwon, Milaniv{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 $kgeq 3$. 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.

Problem

• Bounding tree-independence number in K1d-free graphs
• Proving conjecture for graphs excluding induced wheel minors
• Enabling tractable NP-hard problems on specific graph classes

Innovation

• Generalized brambles for tree-independence number
• Polynomial-time algorithm for induced minors
• Bounded tree-independence number for K1d-free graphs