Directed Metric Dimension of Oriented Graphs with Cyclic Covering

TL;DR

This paper investigates the directed metric dimension of strongly connected oriented graphs with cyclic covering, focusing on wheels, fans, and cycle amalgamations with $C_n$-simple orientations.

Contribution

It introduces the concept of directed metric dimension for graphs with cyclic coverings and analyzes it for specific classes of oriented graphs such as wheels, fans, and cycle amalgamations.

Findings

Determined the directed metric dimension for oriented wheels.

Established the metric dimension for oriented fans.

Analyzed the metric dimension of cycle amalgamations.

Abstract

Let $D$ be a strongly connected oriented graph with vertex-set $V$ and arc-set $A$. The distance from a vertex $u$ to another vertex $v$, $d(u,v)$ is the minimum length of oriented paths from $u$ to $v$. Suppose $B=\{b_1,b_2,b_3,...b_k\}$ is a nonempty ordered subset of $V$. The representation of a vertex $v$ with respect to $B$, $r(v|B)$, is defined as a vector $(d(v,b_1), d(v,b_2), ..., d(v,b_k))$. If any two distinct vertices $u,v$ satisfy $r(u|B)\neq r(v|B)$, then $B$ is said to be a resolving set of $D$. If the cardinality of $B$ is minimum then $B$ is said to be a basis of $D$ and the cardinality of $B$ is called the directed metric dimension of $D$. Let $G$ be the underlying graph of $D$ admitting a $C_n$-covering. A $C_n$-simple orientation is an orientation on $G$ such that every $C_n$ in $D$ is strongly connected. This paper deals with metric dimensions of oriented wheels, oriented fans, and amalgamation of oriented cycles, all of which admitting $C_n$-simple orientations.