Directed graphs and its Boundary Vertices

TL;DR

This paper investigates the metric properties of boundary, contour, eccentric, and peripheral sets in large strongly connected directed graphs, using prime factor decomposition related to the Cartesian product.

Contribution

It introduces a method to identify these metric sets in strong digraphs through their prime factor decomposition with respect to Cartesian product.

Findings

Characterization of boundary and eccentric sets in terms of prime factors.

Decomposition approach simplifies analysis of large strong digraphs.

Provides a framework for understanding metric properties in directed graphs.

Abstract

Suppose that $D=(V,E)$ is a strongly connected digraph. Let $u,v\in V(D)$. The maximum distance $md (u,v)$ is defined as $md(u,v)$=max\{$\overrightarrow{d}(u,v), \overrightarrow{d}(v,u)$\} where $\overrightarrow{d}(u,v)$ denote the length of a shortest directed $u-v$ path in $D$. This is a metric. The boundary, contour, eccentric and peripheral sets of a strong digraph $D$ are defined with respect to this metric. The main aim of this paper is to identify the above said metrically defined sets of a large strong digraph $D$ in terms of its prime factor decomposition with respect to cartesian product.