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

TL;DR

This paper investigates boundary vertices in strongly connected directed graphs based on the sum metric, exploring their relationships and properties, especially in the context of corona product graphs.

Contribution

It introduces new insights into boundary vertices with respect to the sum metric and characterizes the center and boundary sets of corona product digraphs.

Findings

Characterization of boundary vertices in strongly connected digraphs

Relationships among boundary, contour, eccentric, and peripheral vertices

Center determination of corona product of strongly connected digraphs

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.