**Observation 2.**

(a) *If ∂*(*G*) *equals the set of simplicial vertices of G, then GSR* ∼= *<sup>K</sup>*|*∂*(*G*)|*. In particular,* (*Kn*)*SR* ∼= *Kn and for any tree T, TSR* ∼= *Kl*(*T*)*. n*


In [9], realization and characterization problems of the strong resolving graph of a graph as a graph transformation were firstly dealt with. That is, the following problems were studied.


For instance, in [9] was proved that complete graphs, paths and cycles of order larger than four are realizable as the strong resolving graph of other graphs. On the other hand, it was also proved in [9] that stars and cycles of order four are not realizable as strong resolving graphs. Based on these two facts, a conjecture concerning the not realization of complete bipartite graphs in general was pointed out. Such conjecture was recently shown in [10].

In connection with these comments, it would be desirable to continue obtaining some realization (and also characterization - although much more complicated) results for the strong resolving graphs of graphs. We are then aimed in this work to present some realization results which are involving cactus graphs.

### *1.3. Strong Metric Dimension of G versus Vertex Cover Number of GSR*

Oellermann and Peters-Fransen [7] showed that the problem of finding the strong metric dimension of graphs can be transformed into the well-known problem regarding the vertex cover of graphs. A set *S* of vertices of *G* is a *vertex cover* of *G* if every edge of *G* is incident with at least one vertex of *S*. The *vertex cover number* of *G*, denoted by *β*(*G*), is the smallest cardinality of a vertex cover of *G*. We refer to a *β*(*G*)-set in a graph *G* as a vertex cover set of cardinality *β*(*G*).

**Theorem 1** ([7])**.** *For any connected graph G,*

$$\dim\_s(G) = \beta(G\_{SR}).$$

Recall that the largest cardinality of a set of vertices of *G*, no two of which are adjacent, is called the *independence number* of *G* and is denoted by *<sup>α</sup>*(*G*). We refer to an *α*(*G*)-set in a graph *G* as an independent set of cardinality *<sup>α</sup>*(*G*). The following well-known and useful result, due to Gallai, states the relationship between the independence number and the vertex cover number of a graph.

**Theorem 2** (Gallai's theorem)**.** *For any graph G of order n,*

$$
\alpha(G) + \beta(G) = n.
$$

Thus, by using Theorems 1 and 2 we immediately obtain the next result.

**Corollary 1.** *For any graph G,*

$$\dim\_{\mathfrak{s}}(\mathcal{G}) = |\partial(\mathcal{G})| - \mathfrak{a}(\mathcal{G}\_{SR}).$$

### **2. Cactus Graphs: General Issues**

A *cactus graph* (also called a cactus tree) is a connected graph in which any two simple cycles have at most one vertex in common. Equivalently, every edge of the graph belongs to at most one simple cycle. Next we study the strong metric dimension of cactus graphs, and we first give some necessary terminology. Note that a cycle of two vertices is precisely a path on two vertices. A vertex belonging to at least two simple cycles is a *cut vertex*. A cycle having only one cut vertex is called a *terminal cycle*. In a terminal cycle *A*, every vertex being diametral, in the subgraph induced by *A*, with respect to the cut vertex of *A* is a *terminal vertex*. From now on, *τ*(*G*) denotes the set of terminal vertices of *G*. Also, *<sup>ς</sup>*2(*G*) denotes the set of vertices *v*, of degree two, belonging to a cycle of order larger than two, being MMD only with vertices of the same cycle which *v* belongs. Moreover, *<sup>ι</sup>*2(*G*) denotes the set of vertices *u*, of degree two, belonging to a cycle of order larger than two being MMD with at least one vertex of a different cycle which *u* belongs. The following remark can be easily observed.

**Remark 3.** *Let G be a cactus graph. Then, two vertices x*, *y are MMD in G if and only if x*, *y* ∈ *<sup>ς</sup>*2(*G*) ∪ *<sup>ι</sup>*2(*G*) ∪ *<sup>τ</sup>*(*G*)*.*

**Corollary 2.** *For any cactus graph G, ∂*(*G*) = *<sup>ς</sup>*2(*G*) ∪ *<sup>ι</sup>*2(*G*) ∪ *<sup>τ</sup>*(*G*)*.*

**Theorem 3.** *Let G be a cactus graph. Then*

$$|\tau(G)| + \left\lfloor \frac{|\varsigma\_2(G)|}{2} \right\rfloor - 1 \le \dim\_s(G) \le |\tau(G)| + |\iota\_2(G)| + \left\lfloor \frac{|\varsigma\_2(G)|}{2} \right\rfloor.$$

**Proof.** The lower bound follows from the following facts. Any two terminal vertices of *G* are MMD on *G*, and thus, they induce a complete graph of order |*τ*(*G*)|. Also, vertices of *<sup>ς</sup>*2(*G*) induce at least a graph with ) |*<sup>ς</sup>*2(*G*)| 2 \* independent edges that need to be covered in *GSR*. Thus, one needs at least |*τ*(*G*)| − 1 + ) |*<sup>ς</sup>*2(*G*)| 2 \* to strongly resolve all the vertices of *G*.

To see the upper bound, it is only necessary to observe that the set *τ*(*G*) ∪ *<sup>ι</sup>*2(*G*) together with half of vertices of the set *<sup>ς</sup>*2(*G*) form a strong resolving set of *G*, and so, we are done.

Despite the fact that the bounds above are easily proved, we might notice that the problem of describing the strong resolving graph, and similarly, of computing the strong metric dimension of cactus graphs seems to be very challenging based on the situation that we can not control things like the orders of the involved cycles, the number of terminal vertices and cut vertices, their adjacencies, etc. In this sense, it is desirable to introduce extra conditions on the cactus graphs to have more possibilities to give some practical results.

### **3. Strong Resolving Graphs**

In this section we aim to describe the structure of the strong resolving graphs of several different families of cactus graphs. We specifically center our attention into unicyclic graphs, bouquet of cycles and chains of even cycles. With some of these results we contribute to the problem of realization of some graphs as strong resolving graphs, that is, to the problems previously presented.
