Efficient Recognition and Analysis of Thick Forests, a Class of Perfect Graphs

Core Concepts

Thick forests are a class of perfect graphs that can be recognized in polynomial time, unlike most other classes of thick graphs. The author develops efficient algorithms for recognizing thick forests and analyzing their properties, such as counting independent sets and colorings.

Abstract

The content discusses the concept of "thick graphs", where vertices are replaced by cliques and edges are replaced by cobipartite graphs. The author focuses on the class of thick forests, which are a subclass of perfect graphs.
Key highlights:

Thick forests can be recognized in polynomial time, unlike most other classes of thick graphs which are NP-complete.
Thick forests are perfect graphs and resemble chordal graphs, but not all chordal graphs are thick forests.
The author introduces the class of "quasi thick forests", which includes both chordal graphs and thick forests, and shows that this class is a subclass of long hole-free perfect graphs.
The author develops efficient algorithms for counting independent sets and colorings in quasi thick forests, which are #P-complete for general perfect graphs.
The author also considers extensions to larger classes of thick graphs, such as thick triangle-free graphs and thick bounded-treewidth graphs.

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Key Insights Distilled From

Thick Forests

by Mart... at arxiv.org 04-10-2024

https://arxiv.org/pdf/2309.01482.pdf

Deeper Inquiries

What are the potential applications of the efficient recognition and analysis of thick forests in real-world problems

The efficient recognition and analysis of thick forests have several potential applications in real-world problems. One application could be in network analysis, where identifying and understanding the structural properties of thick forests can help in optimizing network connectivity and identifying critical nodes or edges. In social network analysis, thick forests could be used to identify communities or clusters within a network, providing insights into group dynamics and interactions. Additionally, in bioinformatics, thick forests could be utilized to analyze genetic networks or protein interactions, aiding in the understanding of complex biological systems.

How do the structural properties of thick forests compare to other subclasses of perfect graphs, and what insights can be gained from these comparisons

The structural properties of thick forests set them apart from other subclasses of perfect graphs. While perfect graphs without long holes are a subset of thick forests, the unique properties of thick forests, such as the presence of thick vertices and thick edges, distinguish them from classes like chordal graphs or bipartite graphs. These comparisons provide insights into the complexity and richness of graph structures, highlighting the diverse nature of perfect graphs and their subclasses. Understanding these structural differences can lead to advancements in graph theory and algorithm development.

Are there any other interesting graph classes that could be defined using the thick graph construction, and what would be the complexity of recognizing and analyzing such classes

The thick graph construction opens up possibilities for defining and analyzing various interesting graph classes. One potential class could be "thick triangle-free graphs," where the thin graph is triangle-free but the thick graph allows for the presence of thick triangles. Recognizing and analyzing such classes could pose interesting challenges, especially in understanding the interplay between thin and thick structures. The complexity of recognizing and analyzing these classes would depend on the specific properties and constraints imposed, requiring tailored algorithms and approaches for efficient computation.