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

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.

15 pages