Boundary-type Sets of Strong Product of Directed Graphs

TL;DR

This paper investigates boundary-type sets in strong digraphs using the maximum distance metric, focusing on their properties in relation to prime factor decomposition with respect to strong product.

Contribution

It extends the study of boundary-type sets from Cartesian to strong product decompositions in strongly connected digraphs.

Findings

Characterization of boundary sets in strong product digraphs

Relations between boundary sets and prime factor decomposition

Insights into metric properties of strong digraphs

Abstract

Let $D=(V,E)$ be a strongly connected digraph and 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 $D$ with respect to this metric have been defined, and the above said metrically defined sets of a large strong digraph $D$ have been investigated in terms of the factors in its prime factor decomposition with respect to Cartesian product. In this paper we investigate about the above boundary-type sets of a strong digraph $D$ in terms of the factors in its prime factor decomposition with respect to strong product.