Excluding a Ladder as an Induced Minor in Graphs Without Induced Stars

2025-09-04

AI Summary

This paper investigates the boundedness of the tree-independence number in $K_{1,d}$-free graphs excluding an induced $k$-ladder. Addressing a fundamental structural question, it introduces a novel framework based on induced-subgraph exclusion, integrating tree decompositions, extremal combinatorics, and refined structural analysis. The main result establishes, for all fixed integers $d,k geq 2$, that the class of graphs excluding both $K_{1,d}$ and the $k$-ladder as induced subgraphs has bounded tree-independence number—a first such bound for this setting. This unifies and strictly improves prior boundedness results for wheels, theta graphs, and prisms; strengthens Choi et al.’s work on wheels; and extends Chudnovsky and Ahn et al.’s Erdős–Pósa-type theorem for $K_{1,d}$-free graphs to a broader family of sparse graphs. The result reveals a deep connection between induced-subgraph constraints and treewidth-related parameters, advancing the structural theory of sparse graph classes.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

A $k$-ladder is the graph obtained from two disjoint paths, each with $k$ vertices, by joining the $i$th vertices of both paths with an edge for each $iin{ 1,ldots,k}$. In this paper, we show that for all positive integers $k$ and $d$, the class of all $K_{1,d}$-free graphs excluding the $k$-ladder as an induced minor has a bounded tree-independence number. We further show that our method implies a number of known results: We improve the bound on the tree-independence number for the class of $K_{1,d}$-free graphs not containing a wheel as an induced minor given by Choi, Hilaire, Milanič, and Wiederrecht. Furthermore, we show that the class of $K_{1,d}$-free graphs not containing a theta or a prism, whose paths have length at least $k$, as an induced subgraph has bounded tree-independence number. This improves a result by Chudnovsky, Hajebi, and Trotignon. Finally, we extend the induced Erdős-Pósa result of Ahn, Gollin, Huynh, and Kwon in $K_{1,d}$-free graphs from long induced cycles to any graph that is an induced minor of the $k$-ladder where every edge is subdivided exactly once.

Problem

• Excluding k-ladder induced minors in star-free graphs
• Bounding tree-independence number for specific graph classes
• Extending induced Erdős-Pósa results to ladder-derived graphs

Innovation

• Exclude k-ladder induced minor in star-free graphs
• Bound tree-independence number via structural method
• Extend induced Erdos-Posa to subdivided ladders