Core Concepts

The authors present a succinct data structure for chordal graphs with bounded vertex leafage that supports efficient adjacency and neighborhood queries.

Abstract

The key highlights and insights of the content are:
The authors define the class of k-vertex leafage chordal graphs, denoted Gk, which consists of all chordal graphs with vertex leafage at most k and unbounded leafage.
They prove a lower bound of ((k-1)n log n - kn log k - O(log n)) bits on the size of any data structure that encodes a graph in Gk, for k > 0 in o(n/ log n).
For every k-vertex leafage chordal graph G such that k > 1 in o(nc), c > 0, the authors present a ((k-1)n log n + o(kn log n))-bit succinct data structure.
The succinct data structure is constructed by decomposing the sub-trees of the chordal graph's clique tree into paths and using the succinct data structure for path graphs as a black box.
The data structure supports adjacency queries in O(k log n) time and neighborhood queries in O(k^2 d_v log n + log^2 n) time, where d_v is the degree of the vertex v.
The key ideas behind this work are: (1) an improved information theoretic lower bound for chordal graphs with bounded vertex leafage, and (2) a careful decomposition of the sub-trees into paths to leverage the succinct data structure for path graphs.

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by Girish Balak... at **arxiv.org** 04-12-2024

Deeper Inquiries

The succinct data structure for k-vertex leafage chordal graphs has various potential applications in the field of graph theory and algorithm design. Some of the applications include:
Network Analysis: The data structure can be used to efficiently analyze and process large networks represented as chordal graphs with bounded vertex leafage. This can be beneficial in various network-related applications such as social network analysis, communication networks, and biological networks.
Graph Algorithms: The succinct representation can enhance the performance of graph algorithms that operate on chordal graphs. Tasks like shortest path calculations, connectivity analysis, and graph traversals can be optimized using this data structure.
Data Compression: The succinct data structure allows for compact representation of chordal graphs with bounded vertex leafage, enabling efficient storage and transmission of graph data in memory-constrained environments.
Pattern Recognition: The structure can be utilized in pattern recognition tasks where graphs are used to represent complex relationships. By efficiently storing and querying chordal graphs, pattern matching and similarity detection algorithms can be improved.
Bioinformatics: In bioinformatics, where graphs are used to model biological systems and interactions, the succinct data structure can aid in analyzing biological networks, genetic pathways, and protein interactions.

The techniques employed in developing the succinct data structure for k-vertex leafage chordal graphs can be extended to other graph classes beyond chordal graphs by adapting the methodology to suit the specific characteristics of those graph classes. Here are some ways to extend these techniques:
Parameter Generalization: The parameters of vertex leafage and leafage can be generalized to accommodate the properties of different graph classes. By defining suitable parameters for other graph classes, similar succinct data structures can be developed.
Graph Decomposition: The approach of decomposing the graph into substructures, such as paths, trees, or cliques, can be applied to other graph classes with specific structural properties. This decomposition can help in designing efficient data structures for those classes.
Hierarchical Structures: Extending the concept of hierarchical structures, like the clique tree used in chordal graphs, to other graph classes can facilitate the development of succinct representations. Identifying key structural components and their relationships is crucial in this extension.
Optimized Query Operations: Adapting the query operations, such as adjacency queries and neighborhood queries, to suit the characteristics of different graph classes is essential. Tailoring these operations to the specific properties of the graphs can enhance the efficiency of the data structure.

The parameters of vertex leafage and leafage can indeed be generalized to develop succinct data structures for a broader class of graphs beyond chordal graphs. Here's how this generalization can be approached:
Parameter Definition: Define suitable parameters that capture the structural characteristics of the target graph class. These parameters should reflect the key features of the graphs that impact their representation and analysis.
Data Structure Design: Develop a data structure that leverages the generalized parameters to succinctly represent the graphs. Consider the specific requirements and properties of the graph class to optimize the data structure design.
Query Support: Ensure that the data structure supports essential query operations tailored to the generalized parameters. Efficient adjacency queries, neighborhood queries, and other relevant operations should be incorporated based on the graph class's characteristics.
Performance Analysis: Evaluate the performance of the generalized data structure on various graph instances from the target class. Assess its efficiency in terms of space complexity, query processing time, and overall usability.
By extending the concepts of vertex leafage and leafage to a broader range of graph classes and designing corresponding succinct data structures, it is possible to enhance the analysis and manipulation of diverse types of graphs in a space-efficient manner.

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