# Computing the Metric Dimension of Gear Graphs

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

_{2n,m}be a m-level gear graph obtained by m-level wheel graph W

_{2n,m}≅ mC

_{2n}+ k

_{1}by alternatively deleting n spokes of each copy of C

_{2n}and J

_{3n}be a generalized gear graph obtained by alternately deleting 2n spokes of the wheel graph W

_{3n}. In this paper, the metric dimension of certain gear graphs J

_{2n,m}and J

_{3n}generated by wheel has been computed. Also this study extends the previous result given by Tomescu et al. in 2007.

## 1. Introduction and Preliminary Results

## 2. The Metric Dimension of Double Gear Graph J_{2n,m}

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

#### Construction and Observations

- When $n=2,$$dim\left({J}_{4,2}\right)=3+2$, (central vertex v with one major and minor vertex of each ${C}_{2n,i},\text{}1\le i\le 2$ form basis).
- When $n=3,$$dim\left({J}_{6,2}\right)=3+2$, (central vertex v with two minor vertices of each ${C}_{2n,i},1\le i\le 2$ form basis).
- When $n=4,$$dim\left({J}_{8,2}\right)=2+3$, (two minor vertices ${u}^{1},{w}^{1}$ such that $d({u}^{1},{w}^{1})=2$ of ${C}_{2n,1}$ with one minor vertex ${u}^{2}$ and two major vertices ${w}^{2},{x}^{2}$ of ${C}_{2n,2}$ such that $d({u}^{2},{w}^{2})=d({u}^{2},{x}^{2})=3$ form basis).
- When $n=5,$$dim\left({J}_{10,2}\right)=3+4$, (three minor vertices ${u}^{1},{w}^{1},{x}^{1}$ satisfying $d({u}^{1},{w}^{1})=d({w}^{1},{x}^{1})=2$, $d({u}^{1},{x}^{1})=4$ of ${C}_{2n,1}$ with three minor vertices ${u}^{2},{w}^{2},{x}^{2}$ and one major vertex ${z}^{2}$ of ${C}_{2n,2}$ satisfying $d({u}^{2},{w}^{2})=d({w}^{2},{x}^{2})=2$, $d({u}^{2},{x}^{2})=4$ and $d({u}^{2},{z}^{2})=d({w}^{2},{z}^{2})=d({x}^{2},{z}^{2})=3$ form metric basis of ${J}_{10,2}$ ).

- If B is a basis of ${J}_{2n,1}$, $n\ge 6$ then every $2-2$ gap, $2-3$ gap and $3-3$ gap of B contains at most 5, 4 and 3 vertices respectively.
- If B is a basis of ${J}_{2n,1}$, $n\ge 6$ then it contains at most one major gap.
- If B is a basis of ${J}_{2n,1}$, $n\ge 6$ then any two neighboring gaps contain together at most six vertices in which one gap is a major gap.
- If B is a basis of ${J}_{2n,1}$, $n\ge 6$ then any two minor neighboring gaps contain together at most four vertices.

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Case-(i)**: When both gaps are $3-3$ then we have two distinct paths consisting consecutive vertices ${u}_{1},{u}_{2}^{\ast},{u}_{3}$ and ${w}_{1},{w}_{2}^{\ast},{w}_{3}$ of ${C}_{2n,1}$ and ${C}_{2n,2}$ respectively in this case $r\left({u}_{3}^{\ast}\right|B)=r\left({w}_{3}^{\ast}\right|B)$; a contradiction.

**Case-(ii)**: When both gaps are $2-2$ then we have two distinct paths consisting of consecutive vertices ${u}_{1}^{\ast},{u}_{2},{u}_{3}^{\ast},{u}_{4},{u}_{5}^{\ast}$ and ${w}_{1}^{\ast},{w}_{2},{w}_{3}^{\ast},{w}_{4},{w}_{5}^{\ast}$ of ${C}_{2n,1}$ and ${C}_{2n,2}$ respectively but $r\left({u}_{3}^{\ast}\right|B)=r\left({w}_{3}^{\ast}\right|B)$; a contradiction.

**Case-(iii)**: When both gaps are $2-3$ then we have two distinct paths consisting of consecutive vertices ${u}_{1}^{\ast},{u}_{2},{u}_{3}^{\ast},{u}_{4}$ and ${w}_{1}^{\ast},{w}_{2},{w}_{3}^{\ast},{w}_{4}$ of ${C}_{2n,1}$ and ${C}_{2n,2}$ respectively in this case $r\left({u}_{3}^{\ast}\right|B)=r\left({w}_{3}^{\ast}\right|B)$; a contradiction.

**Case-(iv)**: When one gap is $3-3$ and other is $2-2$ gap then we have two distinct paths ${u}_{1},{u}_{2}^{\ast},{u}_{3}$ and ${w}_{1}^{\ast},{w}_{2},{w}_{3}^{\ast},{w}_{4},{w}_{5}^{\ast}$ induced by ${C}_{2n,1}$ and ${C}_{2n,2}$ respectively but $r\left({u}_{2}^{\ast}\right|B)=r\left({w}_{3}^{\ast}\right|B)$; a contradiction.

**Case-(v)**: When one gap is $3-3$ and other is $2-3$ gap then we have two distinct paths consisting of consecutive vertices ${u}_{1},{u}_{2}^{\ast},{u}_{3}$ and ${w}_{1}^{\ast},{w}_{2},{w}_{3}^{\ast},{w}_{4}$ of ${C}_{2n,1}$ and ${C}_{2n,2}$ respectively but $r\left({u}_{2}^{\ast}\right|B)=r\left({w}_{3}^{\ast}\right|B)$; a contradiction.

**Case-(vi)**: When one gap is $2-2$ and other is $2-3$ gap then we have two distinct paths consisting of consecutive vertices ${u}_{1}^{\ast},{u}_{2},{u}_{3}^{\ast},{u}_{4},{u}_{5}^{\ast}$ and ${w}_{1}^{\ast},{w}_{2},{w}_{3}^{\ast},{w}_{4}$ of ${C}_{2n,1}$ and ${C}_{2n,2}$ respectively in this case $r\left({u}_{3}^{\ast}\right|B)=r\left({w}_{3}^{\ast}\right|B)$; a contradiction.

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Case-(i)**: $2-2$ minor gap with three vertices cannot be neighboring gap of $2-2$ minor gap having three vertices, otherwise we have a path consisting of consecutive vertices of ${C}_{2n,1}$ or ${C}_{2n,2}$:${u}_{1}^{\ast},{u}_{2},{u}_{3}^{\ast},{u}_{4},{u}_{5}^{\ast},{u}_{6},{u}_{7}^{\ast}$, where ${u}_{4}\in B$ in this case $r\left({u}_{3}^{\ast}\right|B)=r\left({u}_{5}^{\ast}\right|B)$.

**Case-(ii)**: $2-2$ minor gap with three vertices cannot be neighboring gap of $2-3$ minor gap having two vertices, otherwise we have a path consisting of consecutive vertices of ${C}_{2n,1}$ or ${C}_{2n,2}$:${w}_{1}^{\ast},{w}_{2},{w}_{3}^{\ast},{w}_{4},{w}_{5}^{\ast},{w}_{6}$ where ${w}_{4}\in B$ in this case $r\left({w}_{3}^{\ast}\right|B)=r\left({w}_{5}^{\ast}\right|B)$; a contradiction. □

**Theorem**

**1.**

**Proof.**

**Case-(i)**: When $n\equiv 0\left(mod3\right)$, then we may write $2n=3k$, where $k\ge 4$, is even and $dim\left({J}_{2n,1}\right)+\u2308\frac{2n}{3}\u2309=2k$, in this case W can be considered as:

**Case-(ii)**: When $n\equiv 1\left(mod3\right)$, then we may write $2n=3k+2$, where $k\ge 4$ is even and $dim\left({J}_{2n,1}\right)+\u2308\frac{2n}{3}\u2309=2k+1$, in this case W can be considered as:

**Case-(iii)**: When $n\equiv 2\left(mod3\right)$, then we may write $2n=3k+1$, where $k\ge 5$ is even and $dim\left({J}_{2n,1}\right)+\u2308\frac{2n}{3}\u2309=2k+1$, in this case W can be considered as:

**Case-(i)**: When $r\equiv 0\left(mod2\right)$: In this case $\u230a\frac{r}{2}\u230b=\u2308\frac{r}{2}\u2309=\frac{r}{2}$

**Case-(ii)**: When $r\equiv 1\left(mod2\right)$: In this case $\u230a\frac{r}{2}\u230b=\frac{r-1}{2}$ and $\u2308\frac{r}{2}\u2309=\frac{r+1}{2}$

**Theorem**

**2.**

**Proof.**

## 3. The Metric Dimension of Generalized Gear Graph ${J}_{3n}$

**Definition**

**5.**

#### Construction and Observations

- When $n=2$$dim\left({J}_{6}\right)=2$, (one minor vertex of ${C}_{6}$ together with central vertex v form basis).
- When $n=3$$dim\left({J}_{9}\right)=2=dim\left({J}_{12}\right)$, (two minor vertices ${w}_{1}$ and ${w}_{2}$ such that $d({w}_{1},{w}_{2})=3$ form basis).
- When $n=5$$dim\left({J}_{15}\right)=3$, (three minor vertices ${w}_{1}$,${w}_{2}$ and ${w}_{3}$ such that $d({w}_{1},{w}_{2})=d({w}_{2},{w}_{3})=d({w}_{3},{w}_{4})=4$ form basis).

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

**Case-(i)**: When both gaps are $3-3$ then we have two distinct paths ${u}_{1},{u}_{2},{u}_{3}^{\ast},{u}_{4},{u}_{5}$ and ${w}_{1},{w}_{2},{w}_{3}^{\ast},{w}_{4},{w}_{5}$ but $r\left({u}_{3}^{\ast}\right|B)=r\left({w}_{3}^{\ast}\right|B)$.

**Case-(ii)**: When both gaps are $2-2$ then we have two distinct paths ${u}_{1},{u}_{2}^{\ast},{u}_{3},{u}_{4},{u}_{5}^{\ast},{u}_{6},{u}_{7},{u}_{8}^{\ast}$ and ${w}_{1},{w}_{2}^{\ast},{w}_{3},{w}_{4},{w}_{5}^{\ast},{w}_{6},{w}_{7},{w}_{8}^{\ast}$ but $r\left({u}_{5}^{\ast}\right|B)=r\left({w}_{5}^{\ast}\right|B)$.

**Case-(iii)**: When both gaps are $2-3$ then we have two distinct paths ${u}_{1},{u}_{2}^{\ast},{u}_{3},{u}_{4},{u}_{5}^{\ast},{u}_{6},{u}_{7}$ and ${w}_{1},{w}_{2}^{\ast},{w}_{3},{w}_{4},{w}_{5}^{\ast},{w}_{6},{w}_{7}$ but $r\left({u}_{5}^{\ast}\right|B)=r\left({w}_{5}^{\ast}\right|B)$.

**Case-(iv)**: When one gap is $3-3$ and other is $2-2$ gap then we have two distinct paths ${u}_{1},{u}_{2},{u}_{3}^{\ast},{u}_{4},{u}_{5}$ and ${w}_{1},{w}_{2}^{\ast},{w}_{3},{w}_{4},{w}_{5}^{\ast},{w}_{6},{w}_{7},{w}_{8}^{\ast}$ but $r\left({u}_{3}^{\ast}\right|B)=r\left({w}_{5}^{\ast}\right|B)$.

**Case-(v)**: When one gap is $3-3$ and other is $2-3$ gap then we have two distinct paths ${u}_{1},{u}_{2},{u}_{3}^{\ast},{u}_{4},{u}_{5}$ and ${w}_{1},{w}_{2}^{\ast},{w}_{3},{w}_{4},{w}_{5}^{\ast},{w}_{6},{w}_{7}$ but $r\left({u}_{3}^{\ast}\right|B)=r\left({w}_{5}^{\ast}\right|B)$.

**Case-(vi)**: When one gap is $2-2$ and other is $2-3$ gap then we have two distinct paths ${u}_{1}^{\ast},{u}_{2},{u}_{3},{u}_{4}^{\ast},{u}_{5},{u}_{6},{u}_{7}^{\ast},{u}_{8}$ and ${w}_{1},{w}_{2}^{\ast},{w}_{3},{w}_{4},{w}_{5}^{\ast},{w}_{6},{w}_{7}$ but $r\left({u}_{4}^{\ast}\right|B)=r\left({w}_{5}^{\ast}\right|B)$. □

**Lemma**

**7.**

**Proof.**

**Case-(i)**: When one gap is $2-2$ major gap and the other is $2-2$ minor gap having 7 vertices, then we have two distinct paths ${u}_{1},{u}_{2}^{\ast},{u}_{3},{u}_{4},{u}_{5}^{\ast},{u}_{6},{u}_{7},{u}_{8}^{\ast}$ and ${w}_{1}^{\ast},{w}_{2},{w}_{3},{w}_{4}^{\ast},{w}_{5},{w}_{6},{w}_{7}^{\ast}$ but $r\left({u}_{5}^{\ast}\right|B)=r\left({w}_{4}^{\ast}\right|B)$.

**Case-(ii)**: When one gap is $2-2$ major gap and the other is $2-3$ minor gap having 6 vertices, then we have two distinct paths ${u}_{1},{u}_{2}^{\ast},{u}_{3},{u}_{4},{u}_{5}^{\ast},{u}_{6},{u}_{7},{u}_{8}^{\ast}$ and ${w}_{1}^{\ast},{w}_{2},{w}_{3},{w}_{4}^{\ast},{w}_{5},{w}_{6}$ but $r\left({u}_{5}^{\ast}\right|B)=r\left({w}_{4}^{\ast}\right|B)$.

**Case-(iii)**: When one gap is $2-3$ major gap and the other is $2-2$ minor gap having 7 vertices, then we have two distinct paths ${u}_{1},{u}_{2}^{\ast},{u}_{3},{u}_{4},{u}_{5}^{\ast},{u}_{6},{u}_{7}$ and ${w}_{1}^{\ast},{w}_{2},{w}_{3},{w}_{4}^{\ast},{w}_{5},{w}_{6},{w}_{7}^{\ast}$ but $r\left({u}_{5}^{\ast}\right|B)=r\left({w}_{4}^{\ast}\right|B)$.

**Case-(iv)**: When one gap is $2-3$ major gap and the other is $2-3$ minor gap having 6 vertices, then we have two distinct paths ${u}_{1},{u}_{2}^{\ast},{u}_{3},{u}_{4},{u}_{5}^{\ast},{u}_{6},{u}_{7}$ and ${w}_{1}^{\ast},{w}_{2},{w}_{3},{w}_{4}^{\ast},{w}_{5},{w}_{6}$ but $r\left({u}_{5}^{\ast}\right|B)=r\left({w}_{4}^{\ast}\right|B)$. □

**Lemma**

**8.**

**Proof.**

**Lemma**

**9.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Case-(i)**: When $n\equiv 0\left(mod2\right)$ then W can be considered as:

**Case-(ii)**: When $n\equiv 1\left(mod2\right)$ then W can be considered as:

## 4. Conclusions

**Open Problem.**Determine the metric dimension of m-level generalized gear graph ${J}_{2n,k,m}$.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Tomescu, I.; Javaid, I. On the metric dimension of the jahangir graph. Bull. Math. Soc. Sci. Math. Roum.
**2007**, 50, 371–376. [Google Scholar] - Chartrand, G.; Eroh, L.; Johnson, M.A.; Oellermann, O.R. Resolvability in graphs and metric dimension of a graph. Disc. Appl. Math.
**2000**, 105, 99–113. [Google Scholar] [CrossRef] - Imran, M.; Baig, A.Q.; Bokhary, S.A.; Javaid, I. On the metric dimension of circulant graphs. Appl. Math. Lett.
**2012**, 25, 320–325. [Google Scholar] [CrossRef] - Cameron, P.J.; Van Lint, J.H. Designs, Graphs, Codes and their Links. In London Mathematical Society Student Texts; Cambridge University Press: Cambridge, UK, 1991; Volume 22. [Google Scholar]
- Khuller, S.; Raghavachari, B.; Rosenfeld, A. Localization in Graphs; Technical Report CS-TR-3326; University of Maryland at College Park: College Park, MD, USA, 1994. [Google Scholar]
- Melter, R.A.; Tomescu, I. Metric bases in digital geometry. Graph. Image Process.
**1984**, 25, 113–121. [Google Scholar] [CrossRef] - Slater, P.J. Leaves of trees. Congress
**1975**, 14, 549–559. [Google Scholar] - Harary, F.; Melter, R.A. On the metric dimension of a graph. Ars Combin.
**1976**, 2, 191–195. [Google Scholar] - Tomescu, I.; Imran, M. metric dimension and R-Sets of connected graph. Graphs Combin.
**2011**, 27, 585–591. [Google Scholar] [CrossRef] - Bača, M.; Baskoro, E.T.; Salman, A.N.M.; Saputro, S.W.; Suprijanto, D. On metric dimension of regular bipartite graphs. Bull. Math. Soc. Sci. Math. Roum.
**2011**, 54, 15–28. [Google Scholar] - Buczkowski, P.S.; Chartrand, G.; Poisson, C.; Zhang, P. On k-dimensional graphs and their bases. Perioddica Math. Hung.
**2003**, 46, 9–15. [Google Scholar] [CrossRef] - Siddique, H.M.A.; Imran, M. Computing the metric dimension of wheel related graphs. Appl. Math. Comput.
**2014**, 242, 624–632. [Google Scholar] - Manuel, P.; Rajan, B.; Rajasingh, I.; Monica, C. On minimum metric dimension of honeycomb networks. J. Discret. Algorithms
**2008**, 6, 20–27. [Google Scholar] [CrossRef] - Sebo, A.; Tannier, E. On metric generators of graphs. Math. Oper. Res.
**2004**, 29, 383–393. [Google Scholar] [CrossRef] - Bras, R.L.; Gomes, C.P.; Selman, B. Double-wheel graphs are graceful. In Proceedings of the Twenty-Third International Joint Conference on Artificial Intelligence, Beijing, China, 3–9 August 2013; pp. 587–593. [Google Scholar]

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Imran, S.; Siddiqui, M.K.; Imran, M.; Hussain, M.; Bilal, H.M.; Cheema, I.Z.; Tabraiz, A.; Saleem, Z.
Computing the Metric Dimension of Gear Graphs. *Symmetry* **2018**, *10*, 209.
https://doi.org/10.3390/sym10060209

**AMA Style**

Imran S, Siddiqui MK, Imran M, Hussain M, Bilal HM, Cheema IZ, Tabraiz A, Saleem Z.
Computing the Metric Dimension of Gear Graphs. *Symmetry*. 2018; 10(6):209.
https://doi.org/10.3390/sym10060209

**Chicago/Turabian Style**

Imran, Shahid, Muhammad Kamran Siddiqui, Muhammad Imran, Muhammad Hussain, Hafiz Muhammad Bilal, Imran Zulfiqar Cheema, Ali Tabraiz, and Zeeshan Saleem.
2018. "Computing the Metric Dimension of Gear Graphs" *Symmetry* 10, no. 6: 209.
https://doi.org/10.3390/sym10060209