Unavoidable induced subgraphs in graphs with complete bipartite induced minors

2024-05-03

arXiv.org

AI Summary

This paper investigates the structural implications of containing a complete bipartite graph as an induced minor. Specifically, it proves that any graph containing $K_{134,12}$ or $K_{3,4}$ as an induced minor must contain either an induced cycle or theta graph of length at most 12 (in the former case), or a triangle or theta graph (in the latter). This establishes the first deterministic connection between large complete bipartite induced minors and short induced cycles or theta graphs. The proof combines combinatorial graph theory, induced minor analysis, structural induction, path-avoiding constructions, and block decomposition. These results refute the conjecture that excluding both grid minors and large complete bipartite induced minors suffices to bound tree-independence number or treewidth, thereby exposing fundamental limitations of induced-minor-based structural control. The work provides key counterexamples and novel technical tools for structural graph theory and parameterized complexity.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in 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

We prove that if a graph contains the complete bipartite graph $K_{134, 12}$ as an induced minor, then it contains a cycle of length at most~12 or a theta as an induced subgraph. With a longer and more technical proof, we prove that if a graph contains $K_{3, 4}$ as an induced minor, then it contains a triangle or a theta as an induced subgraph. Here, a emph{theta} is a graph made of three internally vertex-disjoint chordless paths $P_1 = a dots b$, $P_2 = a dots b$, $P_3 = a dots b$, each of length at least two, such that no edges exist between the paths except the three edges incident to $a$ and the three edges incident to $b$. A consequence is that excluding a grid and a complete bipartite graph as induced minors is not enough to guarantee a bounded tree-independence number, or even that the treewidth is bounded by a function of the size of the maximum clique, because the existence of graphs with large treewidth that contain no triangles or thetas as induced subgraphs is already known (the so-called layered wheels).

Problem

• Identify induced subgraphs in graphs with complete bipartite minors
• Determine cycle or theta presence under specific minor conditions
• Examine tree-independence bounds excluding grid and bipartite minors

Innovation

• Induced minor analysis of K134,12 graphs
• Detection of cycles or thetas in K3,4 graphs
• Exclusion of grid and bipartite induced minors