Editing Short Cycles Connectivity: A Generalization of Ordinary Connectivity

Title: Short Cycles Connectivity: A Generalization of Ordinary Connectivity

Abstract:
This research article introduces the concept of short cycles connectivity, a generalization of ordinary connectivity. Instead of being connected by a single path, two vertices are now connected by a sequence of short cycles, where two adjacent cycles share at least one common vertex. The study presents efficient algorithms for determining equivalence classes and discusses further generalizations, such as connectivity by small cliques or other families of graphs. The research suggests that short cycles connectivity can be a powerful tool for analyzing large networks and identifying hierarchical structures.

Main Research Question:
How can we generalize the concept of ordinary connectivity to include short cycles, and what are the implications for network analysis?

Methodology:
The study uses graph theory to define short cycles connectivity, which is a generalization of ordinary connectivity. The method involves creating equivalence classes of vertices based on their ability to be connected by sequences of short cycles. The research presents an algorithm for determining these equivalence classes, which is shown to be efficient for large networks.

Results:
The main result is the development of short cycles connectivity, which is shown to be an equivalence relation on the set of vertices. The study also presents algorithms for determining equivalence classes and discusses further generalizations.

Implications:
The research suggests that short cycles connectivity can be a valuable tool for analyzing large networks. By identifying hierarchical structures and nonhierarchical links, the method can help researchers better understand the complex dynamics of network systems. Additionally, the study's generalizations open up new avenues for further research in graph theory and network analysis.

Link to Article: https://arxiv.org/abs/0308011v2
Authors: 
arXiv ID: 0308011v2

[[Category:Computer Science]]
[[Category:Connectivity]]
[[Category:Cycles]]
[[Category:Short]]
[[Category:Research]]
[[Category:By]]