Graphs Containing K3,4 as an Induced Minor Must Contain a 3-Path Configuration

Conceitos Básicos

If a graph contains K3,4 as an induced minor, then it must contain a 3-path configuration (theta, prism, or pyramid) as an induced subgraph.

Resumo

The key insights and findings of the content are:

The authors prove that if a graph G contains K3,4 as an induced minor, then G must contain a 3-path configuration (theta, prism, or pyramid) as an induced subgraph. This is done through a detailed structural analysis.

As a consequence, the authors show 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. This is 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).

The authors also prove a more general result, showing that if a graph G contains K3,4 as an induced minor, then G must contain a 3-path configuration (theta, prism, or pyramid) as an induced subgraph. This result is best possible in several ways, as discussed in the content.

The proof of the main theorem relies on a precise description of how K3,3 can be contained as an induced minor in a 3PC-free graph, which is of independent interest.

The authors also prove that if a graph G contains a (5x5)-grid as an induced minor, then G must contain a 3-path configuration as an induced subgraph. This result is easier to obtain than the main theorem.

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Principais Insights Extraídos De

Unavoidable induced subgraphs in graphs with complete bipartite induced minors

by Maria Chudno... às arxiv.org 05-06-2024

https://arxiv.org/pdf/2405.01879.pdf

Perguntas Mais Profundas

Are there any other types of induced subgraphs that are guaranteed to appear in graphs containing K3,4 as an induced minor, beyond the 3-path configurations

In graphs containing K3,4 as an induced minor, besides the 3-path configurations guaranteed by Theorem 1.4, there are no other specific induced subgraphs that are guaranteed to appear. The proof of Theorem 1.4 establishes the presence of a 3-path configuration as a direct consequence of the existence of K3,4 as an induced minor. This result is comprehensive and covers all possible induced subgraphs that must exist in such graphs. The structural constraints imposed by the presence of K3,4 as an induced minor lead to the formation of these specific 3-path configurations, making them the only induced subgraphs guaranteed to be present in this context.

Can the result in Theorem 1.4 be extended to other complete bipartite graphs Ka,b as induced minors, or are there fundamental differences in the structural properties of these graphs

The result in Theorem 1.4, which guarantees the existence of a 3-path configuration in graphs containing K3,4 as an induced minor, may not be directly extendable to other complete bipartite graphs Ka,b as induced minors. The structural properties and connectivity constraints imposed by different complete bipartite graphs can lead to distinct induced subgraphs and configurations in the containing graphs. Each complete bipartite graph Ka,b introduces unique connectivity patterns and constraints, which may result in different induced subgraphs being guaranteed to appear in the containing graph. Therefore, it is essential to analyze the specific properties of each complete bipartite graph individually to determine the induced subgraphs that are guaranteed to exist when they are induced minors in a graph.

What are the implications of these results for the design of efficient algorithms for problems like maximum independent set on graph classes with bounded tree-independence number

The results obtained regarding the guaranteed presence of specific induced subgraphs in graphs with complete bipartite induced minors, as discussed in the context provided, have significant implications for the design of efficient algorithms for problems like the maximum independent set on graph classes with bounded tree-independence number. Understanding the structural properties and the induced subgraphs that are forced by the presence of certain induced minors enables algorithm designers to tailor their approaches to exploit these properties effectively. By leveraging the knowledge of the induced subgraphs that must exist in graphs with specific induced minors, algorithmic strategies can be optimized to handle these structures efficiently. This insight can lead to the development of specialized algorithms that take advantage of the known induced subgraphs to improve the computational complexity and performance of solving problems like the maximum independent set on graphs with bounded tree-independence number.