Characterization and Computation of Switching Classes in Graphs

Core Concepts

The authors characterize the maximum subclass and minimum superclass of hereditary graph classes closed under switching, providing insights into recognition algorithms for various graph classes.

Abstract

The content delves into the concept of switching operations in graphs, exploring the characterization and computation of maximum subclasses and minimum superclasses for hereditary graph classes. It discusses polynomial-time algorithms, forbidden induced subgraphs, and implications for various graph categories.
The authors analyze the complexities of recognizing switching classes in different graph categories, highlighting key results such as finite characterizations for specific graph classes and implications for minor-closed graphs. The content provides a comprehensive overview of switching operations in graphs, offering insights into algorithmic recognition and structural properties.

Stats

For several graph classes, we develop polynomial-time algorithms to recognize the minimum superclass.
The lower G switching class is characterized by a finite number of forbidden induced subgraphs when G is certain graph classes.
Let G be a nontrivial minor-closed graph class; the order of every graph in the lower G switching class is smaller than R(p, p(p + 1)/2).
The upper G switching class can be recognized in polynomial time when G is certain graph classes.
Deciding whether a graph is switching equivalent to a P10-free or C7-free graph is NP-Complete.

Quotes

"In a complemen-tary study, Hage and Harju [21] characterized graphs that cannot be switched to any forest."
"Two graphs are switching equivalent if one can be obtained from the other by switching."
"The upper G switching classes turn out to be more complicated."
"Every odd hole of length at least seven contains an induced P4 + K1."
"The lower perfect switching class has only four forbidden induced subgraphs."

Key Insights Distilled From

Switching Classes

by Dhanyamol An... at arxiv.org 03-08-2024

https://arxiv.org/pdf/2403.04263.pdf

Deeper Inquiries

What implications do these findings have on real-world applications involving network optimization

The findings regarding the lower switching classes with an infinite number of forbidden induced subgraphs have significant implications for real-world applications involving network optimization. In practical network design and management scenarios, understanding the constraints imposed by these lower switching classes can help in optimizing network structures to avoid configurations that lead to these forbidden induced subgraphs. By considering the limitations set forth by these lower switching classes, network engineers can make informed decisions to ensure efficient and effective network operations.

How do these results challenge traditional approaches to graph theory problems

These results challenge traditional approaches to graph theory problems by introducing a new perspective on graph properties and their relationships through the concept of switching operations. The idea that certain graph classes have finite versus infinite forbidden induced subgraphs based on their hereditary nature under switching provides a novel way of characterizing graphs beyond conventional methods like minor-closed or hereditary classifications. This challenges researchers and practitioners in graph theory to explore alternative frameworks for analyzing and categorizing graphs based on their behavior under specific operations.

How can the concept of switching operations be applied to other mathematical disciplines beyond graph theory

The concept of switching operations introduced in this context can be applied to other mathematical disciplines beyond graph theory, particularly in areas where transformations or rearrangements play a crucial role. For example, in combinatorial optimization problems, such as permutation-based algorithms or scheduling tasks, understanding how different configurations transform under specific operations akin to switching could lead to innovative problem-solving strategies. Additionally, in algebraic structures or group theory, exploring analogous concepts of "switching" elements within sets could offer insights into symmetry properties or subgroup interactions from a dynamic perspective. Overall, the notion of switching opens up avenues for cross-disciplinary exploration and application across various mathematical domains.