# Remarks on the Vertex and the Edge Metric Dimension of 2-Connected Graphs

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## Abstract

**:**

## 1. Introduction

**Conjecture**

**1.**

**Conjecture**

**2.**

**Conjecture**

**3.**

**Conjecture**

**4.**

**Lemma**

**1.**

## 2. $\Theta $-Graphs with Metric Dimensions Equal to 3

**Lemma**

**2.**

**Proof.**

**Case 1:**${s}_{1}\in V\left({P}_{1}\right)$and${s}_{2}\in V\left({P}_{2}\right)$. Let us denote ${d}_{1}=d({s}_{1},u),$${d}_{2}=d({s}_{2},u),$ $a={d}_{1}+{d}_{2}$, and $b=2p-a$. If $a=b,$ then ${s}_{1}$ and ${s}_{2}$ form an antipodal pair on ${C}_{12}$, which implies that two neighbors of ${s}_{1}$ are not distinguished by S. Therefore, without loss of generality, we may assume $a<p$ and ${d}_{1}\le {d}_{2}$. Since $a+b=2p,$ it follows that a and b are of the same parity; hence, $b-a$ is a positive even number. Therefore, we can define $c=(b-a)/2$, and we know that c is a positive integer. Let $d=2{d}_{1}+c$. Notice that

**Case 2:**${s}_{1}\in V\left({P}_{1}\right)$and${s}_{2}\in V\left({P}_{3}\right)$. For $G={\Theta}_{p,p,p}$, this case is analogous to the previous one, so let us assume $G={\Theta}_{p,p,p+2}$. Again, denote ${d}_{1}=d(u,{s}_{1})$, ${d}_{2}=d(u,{s}_{2}),$$a={d}_{1}+{d}_{2}$, and $b=2p+2-a$. If $a=b,$ then ${s}_{1}$ and ${s}_{2}$ are antipodal on ${C}_{13}$, so the two neighbors of ${s}_{1}$ are not distinguished by S. Hence, without loss of generality, we may assume $a<b.$ Let us denote $c=(b-a)/2.$ Since $a+b=2p+2$ we know that a and b are of the same parity, so $b-a$ is a positive integer. Consequently, also, c is a positive integer.

**Lemma**

**3.**

**Proof.**

**Theorem**

**1.**

**Corollary**

**1.**

**Lemma**

**4.**

**Proof.**

**Proposition**

**1.**

**Proof.**

**Corollary**

**2.**

## 3. $\Theta $-Graphs with Metric Dimensions Equal to 2

**Lemma**

**5.**

- (i)
- If one of p, q, r is odd and at least 3 and one of p, q, r is even, say $q\ge 3$ is odd and r is even, then $S=\{{v}_{(q-1)/2},{w}_{r/2}\}$;
- (ii)
- If $p=1$ and both q and r are even, then $S=\{u,{w}_{r/2}\}$;
- (iii)
- If all p, q, r are even and $q\notin \{p,p+2\},$ then $S=\{{v}_{1},{w}_{r/2}\}$;
- (iv)
- If all of p, q, r are even, $q\in \{p,p+2\}$, and $r\ge p+4,$ then $S=\{{v}_{q/2},{w}_{1}\}$;
- (v)
- If all p, q, r are even and $q=r=p+2,$ then $S=\{{v}_{1},{w}_{1}\}$;
- (vi)
- If all p, q, r are odd and $q\notin \{p,p+2\}$, then $S=\{{v}_{1},{w}_{(r-1)/2}\}$;
- (vii)
- If all p, q, r are odd, $q\in \{p,p+2\}$ and $r\ge p+4$, then $S=\{{v}_{(q-1)/2},{w}_{1}\}$;
- (viii)
- If all p, q, r are odd and $q=r=p+2,$ then $S=\{{v}_{1},{w}_{1}\}$.

**Proof.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**3.**

**Lemma**

**6.**

- (i)
- If $p<q$, $r\ge 3$, and $p+r$ is even, then $S=\{{w}_{(r-p)/2},{w}_{(r+p)/2}\}$;
- (ii)
- If $p<q$, $r\ge p+3$, and $p+r$ is odd, then $S=\{{w}_{\lfloor (r-p)/2\rfloor},{w}_{\lceil (r+p)/2\rceil}\}$;
- (iii)
- If $p<q$, $r=p+1$, and $(p,q,r)\ne (1,2,2)$, then $S=\{{v}_{1},{w}_{1}\}$;
- (iv)
- If $p=q$ and $p\ge 4$, then $S=\{{u}_{2},{v}_{1}\}$;
- (v)
- If $p=q$ and $r\ge p+3$, then $S=\{{v}_{1},{w}_{1}\}$.

**Proof.**

- -
- If $0\le i\le p-5$, then ${d}_{1}=d({u}_{2},{v}_{i+2})=i+4>i+2=d({u}_{2},{w}_{i})={d}_{2}$;
- -
- If $i=p-4$, then ${d}_{1}=d({u}_{2},{v}_{i+3})=p-1>p-2=d({u}_{2},{w}_{i})={d}_{2}$;
- -
- If $i=p-3$, then ${d}_{1}=d({u}_{2},{v}_{i+3})=p-2<p-1=d({u}_{2},{w}_{i})={d}_{2}$.

**Theorem**

**3.**

**Proof.**

**Corollary**

**4.**

## 4. Further Work

**Conjecture**

**5.**

**Conjecture**

**6.**

**Proposition**

**2.**

**Proof.**

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**A set $S=\{{s}_{1},{s}_{2}\}$ in the proof of Lemma 2: (

**a**) case when ${s}_{1}\in V\left({P}_{1}\right)$ and ${s}_{2}\in V\left({P}_{2}\right)$ with $p=6,$${d}_{1}=1,$${d}_{2}=4,$$a=5,$$b=7,$$c=1$, and $d=3$, in which ${u}_{d}$ and ${w}_{c}$ are not distinguished by $S;$ (

**b**) case when ${s}_{1}\in V\left({P}_{1}\right)$ and ${s}_{2}\in V\left({P}_{3}\right)$ with $p=6,$${d}_{1}=3,$${d}_{2}=2,$$a=5,$$b=9,$$c=2$, and $d=8,$ where ${u}_{d}$ and ${v}_{c}$ are not distinguished by $S.$

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**MDPI and ACS Style**

Knor, M.; Sedlar, J.; Škrekovski, R.
Remarks on the Vertex and the Edge Metric Dimension of 2-Connected Graphs. *Mathematics* **2022**, *10*, 2411.
https://doi.org/10.3390/math10142411

**AMA Style**

Knor M, Sedlar J, Škrekovski R.
Remarks on the Vertex and the Edge Metric Dimension of 2-Connected Graphs. *Mathematics*. 2022; 10(14):2411.
https://doi.org/10.3390/math10142411

**Chicago/Turabian Style**

Knor, Martin, Jelena Sedlar, and Riste Škrekovski.
2022. "Remarks on the Vertex and the Edge Metric Dimension of 2-Connected Graphs" *Mathematics* 10, no. 14: 2411.
https://doi.org/10.3390/math10142411