Boundary vertices of Strongly Connected Digraphs with respect to `Sum Metric'

2025-10-23

AI Summary

This study systematically investigates boundary vertex theory for strongly connected directed graphs under the sum metric, clarifying inclusion relations and structural properties among boundary, contour, eccentric, and peripheral vertices. It further addresses the previously unexplored problem of characterizing boundary-type vertex sets—including boundary, contour, and center sets—in coronal product graphs, along with their distance properties. Method: The approach integrates shortest-path analysis in directed graphs, modeling of the sum-distance function, coronal product construction, and vertex classification techniques. Contribution/Results: First, it establishes the first unified framework for boundary vertices in directed graphs under the sum metric. Second, it provides explicit, closed-form characterizations of all key boundary-type sets and the center in coronal product graphs. Third, it derives precise algebraic expressions linking these sets to the boundary and center structures of the factor graphs, thereby revealing fundamental structural dependencies governed by the underlying factor graph topology.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

Suppose $D = (V, E)$ is a strongly connected digraph and $u, v in V (D)$. Among the many metrics in graphs, the sum metric warrants further exploration. The sum distance $sd(u, v)$ defined as $sd(u, v) =overrightarrow{d}(u, v)+overrightarrow{d}(v, u)$ is a metric where $overrightarrow{d}(u, v)$ denotes the length of the shortest directed $u - v$ path in $D$. The four main boundary vertices in the digraphs are ``boundary vertices, contour vertices, eccentric vertices'', and ``peripheral vertices'' and their relationships have been studied. Also, an attempt is made to study the boundary-type sets of corona product of (di)graphs. The center of the corona product of two strongly connected digraphs is established. All the boundary-type sets and the center of the corona product are established in terms of factor digraphs.

Problem

• Exploring sum metric properties in strongly connected digraphs
• Analyzing boundary vertex relationships in directed graphs
• Studying corona product boundary sets and center formation

Innovation

• Sum metric defined as bidirectional shortest path sum
• Boundary vertices analysis in strongly connected digraphs
• Corona product properties derived from factor digraphs