Boundary vertices of Strongly Connected Digraphs with respect to `Sum Metric'
Bijo S. Anand, Manoj Changat, Prasanth G. Narasimha-Shenoi, Mary Shalet Thottungal Joseph, Mithra R, Prakash G. Narasimha-Shenoi
TL;DR
This work studies boundary-type vertex sets in strong digraphs with respect to the sum-distance metric $sd(u,v)=\overrightarrow{d}(u,v)+\overrightarrow{d}(v,u)$ and develops a unified treatment of corona products for both undirected and directed graphs. It first establishes that $sd$ is a metric on strong digraphs, introduces geodesic intervals and geodetic closures, and defines boundary-type sets (boundary, contour, eccentricity, periphery) with key inclusions linking these sets. It then derives explicit distance formulas, radius, diameter, and center results for corona products: for undirected graphs, $r(G\odot H)=r(G)+1$, $d(G\odot H)=d(G)+2$, $C(G\odot H)=C(G)$, with boundary-type sets expressed as unions of copies $H_i$ attached to $G$’s ecc/per vertices; for directed graphs, $\mathrm{rad}(D\odot H)=\mathrm{rad}(D)+2$, $\mathrm{diam}(D\odot H)=\mathrm{diam}(D)+4$, $C(D\odot H)=C(D)$, and boundary-type sets similarly lifted to copies of $H$. This yields a robust framework connecting corona-product boundary structures to those of the factor graphs, with potential applications in network design and metric graph theory.
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.