Directed Metric Dimension of Oriented Graphs with Cyclic Covering

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.