Edge metric dimension of $k$ multiwheel graph

Abstract

If we consider $G=\left(V,E\right)$ to be a connected graph, with $v$ $\in$ $V$ and $e=uw$ $\in$ $E$, then $d_G\left(e,v\right)=min\left\{d_G\left(u,v\right),d_G\left(w,v\right)\right\}$ has been defined as the distance between a vertex $v$ and an edge $e$. The cardinality of the smallest subset $S\subseteq V$ which can assign a unique distance vector to every edge of $G$ is referred to as edge metric dimension (EMD) given by $edim\left(G\right)$. A $k$ multiwheel graph $W_{1,n,k}$ is composed of $k$ cycles $C_n$ along with a central vertex $x$ such that $x$ is adjacent to each of the vertices of $C_1$ and the corresponding vertices of the two consecutive cycles $C_i$ and $C_{i+1}$ are also adjacent for all $1\le i\le \left(k-1\right)$. This article discusses the EMD of the double wheel graph and extends the results to the general $k$ multiwheel graph.