# The Strong Resolving Graph and the Strong Metric Dimension of Cactus Graphs

## Abstract

**:**

## 1. Introduction

#### 1.1. Strong Metric Dimension of Graphs

**Observation**

**1.**

- (a)
- $di{m}_{s}\left(G\right)=n-1$ if and only if $G\cong {K}_{n}$.
- (b)
- If $G\ncong {K}_{n}$, then $di{m}_{s}\left(G\right)\le n-2$.
- (c)
- $di{m}_{s}\left(G\right)=1$ if and only if $G\cong {P}_{n}$.
- (d)
- If $G\cong {C}_{n}$, then $di{m}_{s}\left(G\right)=\lceil n/2\rceil $.
- (e)
- If G is a tree with l leaves, $di{m}_{s}\left(G\right)=l-1$.

**Remark**

**1.**

**Remark**

**2.**

#### 1.2. Strong Resolving Graph of a Graph

**Observation**

**2.**

- (a)
- If $\partial \left(G\right)$ equals the set of simplicial vertices of G, then ${G}_{SR}\cong {K}_{|\partial (G\left)\right|}$. In particular, ${\left({K}_{n}\right)}_{SR}\cong {K}_{n}$ and for any tree T, ${T}_{SR}\cong {K}_{l\left(T\right)}$.
- (b)
- For any 2-antipodal graph G of order n, ${G}_{SR}\cong {\bigcup}_{i=1}^{\frac{n}{2}}{K}_{2}$. In particular, ${\left({C}_{2k}\right)}_{SR}\cong {\bigcup}_{i=1}^{k}{K}_{2}$.
- (c)
- For odd cycles ${\left({C}_{2k+1}\right)}_{SR}\cong {C}_{2k+1}$.
- (d)
- For any complete k-partite graph $G={K}_{{p}_{1},{p}_{2},\dots ,{p}_{k}}$ such that ${p}_{i}\ge 2$, $i\in \{1,2,\dots ,k\}$, ${G}_{SR}\cong {\bigcup}_{i=1}^{k}{K}_{{p}_{i}}$.

**Realization Problem.**Determine which graphs have a given graph as their strong resolving graphs.**Characterization Problem.**Characterize those graphs that are strong resolving graphs of some graphs.

#### 1.3. Strong Metric Dimension of G versus Vertex Cover Number of ${G}_{SR}$

**Theorem**

**1**

**Theorem**

**2**

**.**For any graph G of order n,

**Corollary**

**1.**

## 2. Cactus Graphs: General Issues

**Remark**

**3.**

**Corollary**

**2.**

**Theorem**

**3.**

**Proof.**

## 3. Strong Resolving Graphs

#### 3.1. Unicyclic Graphs

**Remark**

**4.**

**Remark**

**5.**

**Corollary**

**3.**

- w is a terminal vertex of a vertex u of ${C}_{r}$ such that $u,v$ are diametral vertices in ${C}_{r}$.
- w is a diametral vertex with v in ${C}_{r}$ and $w\in {c}_{2}\left(G\right)$.

#### ${G}_{SR}$ for r even.

- The set $T\left(G\right)$ forms a clique in ${G}_{SR}$ and each vertex of $T\left(G\right)$ has at most one neighbor in ${c}_{2}\left(G\right)$.
- If $x,y\in {c}_{2}\left(G\right)$ are diametral vertices in ${C}_{r}$, then $\langle \{x,y\}\rangle $ is a connected component of ${G}_{SR}$ isomorphic to ${K}_{2}$.
- If $x,y$ are diametral vertices in ${C}_{r}$, $x\in {c}_{2}\left(G\right)$ and $y\notin {c}_{2}\left(G\right)$, then $\left\{x\right\}\cup t\left(y\right)$ forms a subgraph of ${G}_{SR}$ isomorphic to ${K}_{\left|t\right(y\left)\right|+1}$ and ${N}_{{G}_{SR}}\left(x\right)=t\left(y\right)$.

#### ${G}_{SR}$ for r odd.

- The set $T\left(G\right)$ forms a clique in ${G}_{SR}$ and each vertex of $T\left(G\right)$ has at most two neighbors in ${c}_{2}\left(G\right)$.
- Let $u\in {c}_{2}\left(G\right)$ and let $x,y$ being diametral vertices with u in ${C}_{r}$.
- –
- If $x,y\in {c}_{2}\left(G\right)$, then $\langle \{u,x,y\}\rangle $ is a subgraph of ${G}_{SR}$ isomorphic to ${P}_{3}$, ${N}_{{G}_{SR}}\left(u\right)=\{x,y\}$ and for every $w\in \{x,y\}$, ${\delta}_{{G}_{SR}}\left(w\right)\ge 2$.
- –
- If $x,y\notin {c}_{2}\left(G\right)$, then $\langle \left\{u\right\}\cup t\left(x\right)\cup t\left(y\right)\rangle $ is a subgraph of ${G}_{SR}$ isomorphic to ${K}_{\left|t\right(x\left)\right|+\left|t\right(y\left)\right|+1}$, ${N}_{{G}_{SR}}\left(u\right)=t\left(x\right)\cup t\left(y\right)$ and for every $w\in t\left(x\right)\cup t\left(y\right)$, ${\delta}_{{G}_{SR}}\left(w\right)\ge \left|t\left(x\right)\right|+\left|t\left(y\right)\right|+1$ for $r\ge 5$ (notice that if $r=3$, then ${\delta}_{{G}_{SR}}\left(w\right)=\left|t\left(x\right)\right|+\left|t\left(y\right)\right|$).
- –
- If $x\in {c}_{2}\left(G\right)$ and $y\notin {c}_{2}\left(G\right)$, then the set $\{u,x\}\cup t\left(y\right)$ form a subgraph (not induced) (Notice that the vertices $t\left(y\right)$ are adjacent between them in ${G}_{SR}$.) of ${G}_{SR}$ isomorphic to a star graph ${S}_{1,\left|t\right(y\left)\right|+1}$ with central vertex u, ${N}_{{G}_{SR}}\left(u\right)=\left\{x\right\}\cup t\left(y\right)$, ${\delta}_{{G}_{SR}}\left(x\right)\ge 2$ and for every $w\in t\left(y\right)$, ${\delta}_{{G}_{SR}}\left(w\right)\ge \left|t\left(y\right)\right|+1$.

**Lemma**

**1.**

**Proof.**

**Proposition**

**1.**

**Proof.**

#### 3.2. Bouquet of Cycles

- The set of $2b$ vertices of the cycles ${C}_{{s}_{1}},{C}_{{s}_{2}},\dots ,{C}_{{s}_{b}}$ which are diametral with w induces a complete multipartite graph ${K}_{2,\dots ,2}$ with b bipartition sets each of cardinality two in ${B}_{SR}$. We denote such set as ${V}_{2b}$ (in Figure 2 and Figure 3, the red colored vertices).
- The set of vertices of each odd cycle ${C}_{{s}_{i}}$, $i\in \{1,\dots ,b\}$, which are different from w induces a path of order ${s}_{i}-1$, in ${B}_{SR}$, whose leaves are the two vertices that are diametral with w.
- The set of vertices of each cycle ${C}_{{r}_{j}}$, $j\in \{1,\dots ,a\}$, which are not diametral with w induces a graph isomorphic to the disjoint union of $({r}_{j}-2)/2$ complete graphs ${K}_{2}$ in ${B}_{SR}$.
- Every three vertices $x,y,z$ such that $x\in {V}_{a}$, $y\in {V}_{2b}$ and $z\in {V}_{2c}$ are pairwise adjacent.

**Proposition**

**2.**

**Proof.**

**Corollary**

**4.**

#### 3.3. Chains of Even Cycles

- We begin with $k+1$ straight chains of even cycles, say ${G}_{0},\dots ,{G}_{k}$, satisfying that the last cycle of the straight chain ${G}_{i}$ is isomorphic to the first cycle of the straight chain ${G}_{i+1}$ for every $i\in \{0,\dots ,k-1\}$.
- Assume that the last cycle of each straight chain ${G}_{i}$ is ${C}_{r}^{i}={v}_{0}^{i}{v}_{1}^{i}\cdots {v}_{r-1}^{i}{v}_{0}^{i}$, for every $i\in \{0,\dots ,k\}$. By the item above, this ${C}_{r}^{i}$ (in ${G}_{i}$) is isomorphic to the first cycle of the straight chain ${G}_{i+1}$ with $i\in \{0,\dots ,k-1\}$.
- Assume also that the terminal vertices of each straight chain ${G}_{i}$ are ${a}_{i},{b}_{k-i}$, for every $i\in \{0,\dots ,k\}$.
- To construct our chain of even cycles $F\in {\mathcal{F}}_{k}$, for every $i\in \{0,\dots ,k-1\}$, we identify the last cycle ${C}_{r}^{i}$ of ${G}_{i}$ with the first cycle ${C}_{r}^{i+1}$ of ${G}_{i+1}$ (that are isomorphic) as follows. Every vertex ${v}_{j}^{i}$ of ${C}_{r}^{i}$ is identified with the vertex ${v}_{j+t}^{i+1}$ for some $t\ne 0$ and every $j\in \{0,r-1\}$ (operations with the subindex of v are done modulo r).

**Remark**

**6.**

**Remark**

**7.**

**Observation**

**3.**

- The terminal vertex ${a}_{i}$ of the straight chain ${G}_{i}$ is MMD with every vertex ${b}_{k-j}$ of the straight chain ${G}_{j}$.
- The terminal vertex ${b}_{i}$ of the straight chain ${G}_{k-i}$ is MMD with every vertex ${a}_{k-j}$ of the straight chain ${G}_{k-j}$.
- In any cycle of F, any pair of diametral (in the cycle) vertices being not cut nor terminal vertices of F are MMD.

- The set of vertices ${a}_{i}$ and ${b}_{i}$, with $i\in \{0,\dots ,k\}$, forms a component of the graph ${F}_{SR}$ isomorphic to a bipartite graph ${J}_{k}$.
- In each cycle of F, each pair of diametral vertices in the cycle, not including terminal nor cut vertices, induces a graph isomorphic to ${K}_{2}$ in ${F}_{SR}$.

**Corollary**

**5.**

## 4. The Strong Metric Dimension

#### 4.1. Unicyclic Graphs

**Lemma**

**2.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Proposition**

**3.**

- (i)
- $\left|t\right(x\left)\right|\le 1$ for every x of ${C}_{r}$.
- (ii)
- There is at most one pair of diametral vertices in ${C}_{r}$ each one having one terminal vertex.

**Proof.**

#### 4.2. Bouquet of Cycles

**Theorem**

**6.**

**Proof.**

#### 4.3. Chains of Even Cycles

**Lemma**

**3.**

**Proof.**

**Theorem**

**7.**

**Proof.**

## 5. Concluding Remarks

- Describe the structure of the strong resolving graphs of some classes of cactus graphs, and compute the strong metric dimension of the graphs in such families.
- Apply the results concerning the descriptions of the strong resolving graphs of the graphs given in the work to other problems, like for instance computing the strong partition dimension (see [4]) of such graphs.
- Continue the lines of this study for other more general families that cactus graphs. This could include for instance, planar graphs or chordal graphs.

## Funding

## Conflicts of Interest

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**Figure 2.**A bouquet of cycles $B\in {\mathcal{B}}_{2,2,1}$ containing the cycles ${C}_{6}$, ${C}_{4}$, ${C}_{9}$, ${C}_{7}$ and ${C}_{3}$.

**Figure 4.**A chain of cycles $F\in {\mathcal{F}}_{3}$ containing six cycles ${C}_{4}$ and two cycles ${C}_{6}$.

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Kuziak, D. The Strong Resolving Graph and the Strong Metric Dimension of Cactus Graphs. *Mathematics* **2020**, *8*, 1266.
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Kuziak D. The Strong Resolving Graph and the Strong Metric Dimension of Cactus Graphs. *Mathematics*. 2020; 8(8):1266.
https://doi.org/10.3390/math8081266

**Chicago/Turabian Style**

Kuziak, Dorota. 2020. "The Strong Resolving Graph and the Strong Metric Dimension of Cactus Graphs" *Mathematics* 8, no. 8: 1266.
https://doi.org/10.3390/math8081266