# On fractional metric dimension of comb product graphs

• Suhadi Wido Saputro Department of Mathematics, Institut Teknologi Bandung, Indonesia
• Andrea Semanicova-Fenovc Department of Applied Mathematics and Informatics, Techical University, Ko\v{s}ice, Slovakia
• Martin Baca
• Marcela Lascsakova
Keywords: Comb Product, Fractional Metric Dimension, Resolving Function

### Abstract

A vertex $z$ in a connected graph $G$ \textit{resolves} two vertices $u$ and $v$ in $G$ if $d_G(u,z)\neq d_G(v,z)$. \ 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.

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Published
2018-02-27
How to Cite
Saputro, S. W., Semanicova-Fenovc, A., Baca, M., & Lascsakova, M. (2018). On fractional metric dimension of comb product graphs. Statistics, Optimization & Information Computing, 6(1), 150-158. https://doi.org/10.19139/soic.v6i1.473
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Section
Research Articles