Efficient Recognition of Chordal and Cograph Cover-Incomparability Graphs

Temel Kavramlar

Chordal graphs having at most two independent simplicial vertices are exactly the chordal graphs which are also cover-incomparability (C-I) graphs. A similar result holds for cographs, and linear-time recognition algorithms are developed for these classes of C-I graphs.

Özet

The content discusses the structure and recognition of chordal and cograph cover-incomparability (C-I) graphs.
Key highlights:

Chordal graphs having at most two independent simplicial vertices are exactly the chordal graphs which are also C-I graphs.
A similar result is obtained for cographs - cographs which are C-I graphs are characterized in terms of their cotree structure.
Linear-time recognition algorithms are developed for chordal C-I graphs and C-I cographs based on the structural characterizations.
C-I graphs are an interesting class of graphs derived from posets, where the edge set is the union of the cover graph and the incomparability graph of the poset.
The recognition complexity of C-I graphs is known to be NP-complete, motivating the study of structural properties and efficient recognition of specific subclasses.
Chordal graphs and cographs are two well-studied graph classes with wide applications, and their intersection with C-I graphs is investigated.

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Önemli Bilgiler Şuradan Elde Edildi

Recognition of chordal graphs and cographs which are Cover-Incomparability graphs

by Arun Anil,Ma... : arxiv.org 04-10-2024

https://arxiv.org/pdf/2307.13964.pdf

Daha Derin Sorular

What are some practical applications of chordal C-I graphs and C-I cographs beyond the theoretical interest

Chordal C-I graphs and C-I cographs have practical applications beyond theoretical interest in various fields. One practical application is in bioinformatics, where these graph structures are used to model biological networks such as protein-protein interaction networks or gene regulatory networks. By representing these networks as chordal C-I graphs or C-I cographs, researchers can analyze the relationships between biological entities efficiently. This analysis can lead to insights into disease mechanisms, drug discovery, and personalized medicine.
Another application is in social network analysis, where chordal C-I graphs and C-I cographs can be used to model relationships between individuals or groups. By studying these graph structures, researchers can identify influential nodes, detect communities, and analyze information flow within the network. This information can be valuable for targeted marketing, recommendation systems, and understanding social dynamics.
Furthermore, in telecommunications and network routing, chordal C-I graphs and C-I cographs can be utilized to optimize network design and routing algorithms. By representing network topologies using these graph structures, engineers can ensure efficient data transmission, fault tolerance, and scalability in communication networks.

How can the structural insights developed in this work be extended to recognize other subclasses of C-I graphs efficiently

The structural insights developed in the study of chordal C-I graphs and C-I cographs can be extended to recognize other subclasses of C-I graphs efficiently by leveraging similar characteristics and properties. For example, the concept of simplicial vertices and the relationship between maximal cliques and universal vertices can be applied to recognize split graphs, which are another subclass of C-I graphs. By identifying key structural features and developing algorithms based on these insights, efficient recognition algorithms can be designed for split graphs and other subclasses of C-I graphs.
Additionally, the linear time recognition algorithms developed for chordal C-I graphs and C-I cographs can serve as a foundation for recognizing related graph classes. By adapting the algorithms and incorporating specific structural properties of different subclasses of C-I graphs, it is possible to extend the recognition framework to cover a broader range of graph classes efficiently.

Are there any connections between the cover-incomparability graphs studied here and other graph classes derived from posets, such as comparability graphs or interval graphs

There are connections between the cover-incomparability graphs studied here and other graph classes derived from posets, such as comparability graphs and interval graphs. Comparability graphs are graphs that can be represented as the intersection graphs of linear orders, while interval graphs are graphs that can be represented as the intersection graphs of intervals on a line.
The relationship between cover-incomparability graphs and comparability graphs lies in their poset representations. Cover-incomparability graphs are derived from posets where the edge set is the union of the cover graph and the incomparability graph. Comparability graphs, on the other hand, are derived from posets where every pair of elements is comparable. By studying the structural differences between these posets, it is possible to identify distinguishing features between cover-incomparability graphs and comparability graphs.
Interval graphs, on the other hand, have a different representation based on intervals on a line. While cover-incomparability graphs focus on the cover and incomparability relationships within a poset, interval graphs capture the intersection relationships between intervals. By exploring the connections between these different graph classes and their poset representations, researchers can gain a deeper understanding of the underlying structures and properties of these graphs.