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 \( B = \{b_1, b_2, b_3, \ldots, b_k\} \) 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 \( (d(v,b_1), d(v,b_2), \ldots, d(v,b_k)) \). 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 admit \( C_n \)-simple orientations.