**1. Introduction**

Topics concerning metric dimension and related parameters in graphs are nowadays very common in the research community, probably based on its applicability to diverse practical problems of identification of nodes in networks. One can find in the literature a large number of works dealing with this topic, both from the applied and theoretical points of views. A popular research line in this subject concerns studying different variants of metric dimension in graphs, which have had their beginnings in the seminal standard metric dimension concept. Some of the most recent ones are probably the edge metric dimension [1], the mixed metric dimension [2], the *k*-metric antidimension [3], the strong partition dimension [4], and the multiset dimension [5,6], just to cite a few recent and remarkable cases. One other interesting version is the strong metric dimension [7], which is now relatively well studied, although a few open questions on this are still open. A fairly complete study on results and open questions concerning the strong metric dimension of graphs can be found in [8].

One significant reason for the interest of several researchers in the strong metric dimension of graphs concerns the closed relationship that exists between such parameter and the very well known vertex cover number of graphs (and thus with the independence number, based on the Gallai's Theorem). To see this relationship, for a given graph *G*, the construction of a new related graph, called strong resolving graph, was required. This graph transformation clearly raised some other related questions on the transformation itself. That is for instance, given a graph *G*: can some properties of the strong resolving graph of *G* be deduced? or; can we realize every graph *H* as the strong resolving graph of another graph *H*? These ones and several other questions were dealt with in [9], which was the first work paying specific attention to the strong resolving graphs of graphs as a special graph transformation. See also [10], where an open problem from [9] was settled.

Clearly, and as we will further notice, a good knowledge of the strong resolving graph of a graph brings important contributions to studying the strong metric dimension of graphs. In this sense, this work is precisely aimed to study the strong resolving graphs and the strong metric dimension of cactus graphs, with some emphasis on different special structures of such cactus graphs. As one will also note through our exposition, strong resolving graphs are very challenging for those graphs having a large number of induced cycles. Thus, cactus graphs represent a significant example of such a situation. With this work, we also contribute to some open problems presented in [9].

The study of the strong metric dimension of some classes of cactus graphs was started in [11,12] where the authors presented some general results for the strong metric dimension of corona product graph and rooted product graphs, respectively. Clear definitions of these two graph products can be found in [8]. A corona product graph or a rooted product graph can have the structure of a cactus graph, depending on which are the graphs used as factors in the product. For instance, if *G* is a cycle and *H* is a graph whose components are only singleton vertices or complete graphs *K*2, then it happens that the corona product graph *G H* is a cactus graphs. To generate a rooted product graph that is a cactus graph, we may consider for example two graphs *G* and *H* which are paths or cycles.

On the other hand, we must mention that the strong metric dimension of unicyclic graphs (which is a cactus graph too) was studied in [13]. There, among other results, several relationships between the strong metric dimension of a unicyclic graph and that of its complement were given. A few other sporadic results can be found in some other articles dealing with related topics that could include examples of cactus graphs. However, we prefer to not include more references that are not essentially connected with this article.

We hence now begin to formalize all the required notations and terminologies that shall be used throughout the document. To this end, for the whole exposition, let *G* be a connected simple graph with vertex set *<sup>V</sup>*(*G*). For two adjacent vertices *x*, *y* ∈ *<sup>V</sup>*(*G*), we use the notation *x* ∼ *y*. For a vertex *x* of *G*, *NG*(*x*) denotes the set of neighbors that *x* has in *G*, i.e., *NG*(*x*) = {*y* ∈ *V*(*G*) : *y* ∼ *<sup>x</sup>*}. The set *NG*(*x*) is called the *open neighborhood of a vertex x* in *G* and *NG*[*x*] = *NG*(*x*) ∪ {*x*} is called the *closed neighborhood of a vertex x* in *G*. The *degree* of the vertex *x* is *<sup>δ</sup>G*(*x*) = |*NG*(*x*)|. The diameter of *G* is defined as *D*(*G*) = *maxx*,*y*∈*<sup>V</sup>*(*G*){*dG*(*<sup>x</sup>*, *y*)}, where *dG*(*<sup>x</sup>*, *y*) is the length of a shortest path between *x* and *y* (a shortest *x*, *y* path). Two vertices *x*, *y* are called *diametral* if *dG*(*<sup>x</sup>*, *y*) = *<sup>D</sup>*(*G*). For a set *S* ⊂ *<sup>V</sup>*(*G*), by *S* we represent the subgraph induced by *S* in *G*.

### *1.1. Strong Metric Dimension of Graphs*

For two distinct vertices *u*, *v* ∈ *<sup>V</sup>*(*G*), a vertex *w* ∈ *V*(*G*) *strongly resolves u*, *v* if there is a shortest *u*, *w* path containing *v*, or a shortest *v*, *w* path containing *u*. Note that it could happen *w* ∈ {*<sup>u</sup>*, *<sup>v</sup>*}. A set *S* of vertices of *G* is a *strong resolving set* for *G*, if every two vertices of *G* are strongly resolved by some vertex of *S*. The smallest cardinality among all strong resolving sets for *G* is called the *strong metric dimension* of *G*, and is denoted by *dims*(*G*). We say that a strong resolving set for *G* of cardinality *dims*(*G*) is a *strong metric basis* of *G*. It next appears the value of the strong metric dimension of some basic graphs.

**Observation 1.** *Let G be a connected graph G of order n* ≥ 2*.*

(a) *dims*(*G*) = *n* − 1 *if and only if G* ∼= *Kn.*

(b) *If G* ∼= *Kn, then dims*(*G*) ≤ *n* − 2*.*


It is said that a vertex *u* of *G* is *maximally distant* from *v* if for every *w* ∈ *NG*(*u*), it happens *dG*(*<sup>v</sup>*, *w*) ≤ *dG*(*<sup>u</sup>*, *<sup>v</sup>*). If *u* is maximally distant from *v* and *v* is maximally distant from *u*, then *u* and *v* are *mutually maximally distant*, and we write that *u*, *v* are MMD in *G*. The set of MMD vertices of *G* is denoted by *∂*(*G*). Note that the set of MMD vertices of a graph *G* is also known as the *boundary* of *G*, as defined in [14,15]. An explanation on the equivalence of these two objects can be readily observed, but also found in [16]. From these definitions, the following remarks are straightforward to observe.

### **Remark 1.** *Let G be a connected graph. Then every two vertices with degree* 1 *are MMD in G.*

For any two mutually maximally distant vertices in *G*, there is no vertex of *G* that strongly resolves them, except themselves. This allows to claim the following.

**Remark 2.** *For every pair of mutually maximally distant vertices x*, *y of a connected graph G, and for every strong metric basis S of G, it follows that x* ∈ *S or y* ∈ *S.*

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

Given a connected graph *G*, the *strong resolving graph* of *G*, denoted by *GSR*, has vertex set *∂*(*G*) and two vertices *u*, *v* are adjacent if and only if *u* and *v* are MMD in *G*. We must remark that the strong resolving graph of a graph *G* was defined in [7] as the graph with vertex set *V*(*G*) and two vertices *u*, *v* are adjacent if and only if *u* and *v* are MMD in *G*. Observe that the difference between these two definitions is the existence of isolated vertices in the strong resolving graph from [7]. The main reason of using in this work the slightly different version is to have a simpler notation and more clarity while proving the results. Moreover, this fact does not influence on the computations we made.

For several basic families of graphs, describing their strong resolving graphs is a straightforward problem. We next recall some examples, which will maybe further useful, and to this end, we recall that a vertex *v* of a graph *G* is *simplicial*, if its closed neighborhood induces a complete graph, and also that a graph *G* is 2-*antipodal* if every vertex of *G* is diametral with exactly one other vertex of *G*.
