Edge-Apexing in Hereditary Classes of Graphs Analysis

Kernekoncepter

Hereditary classes of graphs and their edge-apex properties are explored, focusing on forbidden induced subgraphs.

Resumé

The content delves into hereditary classes of graphs, particularly cographs and edge-apex cographs. It discusses the concept of forbidden induced subgraphs within these classes and how they relate to the overall structure of the graphs. The authors establish that if a hereditary class G has finitely many forbidden induced subgraphs, then so does the edge-apex class Gepex. They characterize edge-apex cographs by bounding the order of forbidden induced subgraphs and finding all of them through computer search. The analysis includes detailed proofs, propositions, and algorithms for identifying forbidden induced subgraphs in edge-apex cographs.

Statistik

Theorem 1.3: Let G be a forbidden induced subgraph for the class of edge-apex cographs. Then 5 ≤ |V (G)| ≤ 8.
Proposition 2.2: If G is a forbidden induced subgraph for Gepex, then |V (G)| ≤ max {2c, c + k(c − 2)}.
Lemma 3.1: Let G be a forbidden induced subgraph for the class of edge-apex cographs and assume that G or G is disconnected. Then G contains two vertex-disjoint P4’s and |V (G)| = 8.

Citater

"We consider finite, simple, undirected graphs."
"A graph G is an edge-apex cograph if G is a cograph or has an edge e such that G − e is a cograph."
"Let abcd and 1234 be two vertex-disjoint induced P4’s in G."

Vigtigste indsigter udtrukket fra

Edge-apexing in hereditary classes of graphs

by Jagdeep Sing... kl. arxiv.org 03-15-2024

https://arxiv.org/pdf/2403.09456.pdf

Dybere Forespørgsler

How do these findings on edge-apexing impact broader graph theory research

The findings on edge-apexing have significant implications for broader graph theory research. By characterizing the forbidden induced subgraphs for edge-apex cographs and establishing bounds on their sizes, this study contributes to a deeper understanding of structural properties within hereditary classes of graphs. These results can potentially lead to the development of new algorithms or methodologies for analyzing and classifying graphs based on their edge-apex properties. Furthermore, insights gained from this research may inspire further investigations into the relationships between different graph classes and their respective forbidden subgraph structures.

What potential limitations or criticisms could arise regarding the approach to identifying forbidden induced subgraphs

One potential limitation or criticism that could arise regarding the approach to identifying forbidden induced subgraphs is related to computational complexity. While the algorithm implemented using SageMath efficiently listed all such subgraphs for small orders, scaling it up to larger graphs might pose challenges in terms of computational resources and time. Additionally, there could be concerns about generalizability when applying these findings beyond specific classes like cographs. It's essential to consider whether similar approaches would yield meaningful results across diverse graph families or if modifications are needed based on varying characteristics.

How might advancements in computational tools like SageMath influence future studies on graph structures

Advancements in computational tools like SageMath play a crucial role in shaping future studies on graph structures by enabling researchers to explore complex problems more effectively. The ability to generate, manipulate, and analyze large sets of graphs allows for comprehensive investigations into various graph properties, including identifying forbidden induced subgraphs as demonstrated in this study. Improved computational tools not only facilitate faster computations but also open up possibilities for exploring more extensive datasets and conducting sophisticated analyses that were previously unattainable manually. As these tools continue to evolve, they will likely drive further advancements in graph theory research by providing robust platforms for experimentation and discovery.