On fractional metric dimension of comb product graphs
Suhadi Wido Saputro, Andrea Semaničová–Feňovčíková, Martin Bača, Marcela Lascśaková · Statistics Optimization & Information Computing · 2018
A vertex $z$ in a connected graph $G$ \textit{resolves} two vertices $u$ and $v$ in $G$ if $d_G(u,z) eq d_G(v,z)$. \ A set of vertices $R_G\{u,v\}$ is a set of all resolving vertices of $u$ and $v$ in $G$. \ For every two distinct vertices $u$ and $v$ in $G$, a \textit{resolving function} $f$ of $G$ is a real function $f:V(G)\rightarrow[0,1]$ such that $f(R_G\{u,v\})\geq1$. \ The minimum value of $f(V(G))$ from all resolving functions $f$ of $G$ is called the \textit{fractional metric dimension} of $G$. \ In this paper, we consider a graph which is obtained by the comb product between two connected graphs $G$ and $H$, denoted by $G\rhd_o H$. \ For any connected graphs $G$, we determine the fractional metric dimension of $G\rhd_o H$ where $H$ is a connected graph having a stem or a major vertex.