**1. Introduction**

Topics concerning distances in graphs are widely studied in the literature, and a high number of applications to real life problems can be found in the literature. As a sporadic example of a work that gives some ideas on the vastness of this topic we cite, for instance [1]. Metric graph theory is a significant area in graph theory that deals with distances in graphs, and a large number of works on this topic is nowadays being developed. One of the lines belonging to metric graph theory is that of the metric dimension parameters. Such topic is indeed a huge area of research that is lastly intensively dealt with. It is then not our goal to enter into citing several articles which are not connected exactly with our exposition. To those readers interested in metric dimension things, we sugges<sup>t</sup> for instance the Ph.D. dissertation [2] (and references cited therein), which contains a good background on the topic.

For any given simple and connected graph *G* whose vertex set is represented as *V*(*G*) and its edge set by *<sup>E</sup>*(*G*), while considering it as a metric space, several styles of metrics over the vertex set *V*, provided with the standard vertex distance, are nowadays defined and studied in the literature.

For instance, the metric *dG* : *V*(*G*) × *V*(*G*) → N ∪ {0}, where N represents the set of positive integers numbers, and *dG*(*<sup>x</sup>*, *y*) is taken as the length of a shortest *u* − *v* path, is one of the most commonly studied. In this sense, the pair (*V*(*G*), *dG*) is clearly a metric space. Concerning such a metric space, it is said that a vertex *v* ∈ *V*(*G*) distinguishes (recognizes or determines are also used terms) two vertices *x* and *y* if *dG*(*<sup>v</sup>*, *x*) = *dG*(*<sup>v</sup>*, *y*). A set *S* ⊂ *V*(*G*) is said to be a *metric generator* for the graph *G* if it is satisfied that any pair of vertices of *G* is uniquely determined by some element of *S*. Consider that *S* = {*<sup>w</sup>*1, *w*2, ... , *wk*} is an ordered subset of vertices of *G*. The *metric vector* (or metric representation) of a given vertex *v* ∈ *<sup>V</sup>*(*G*), with respect to *S*, is the vector of distance (*d*(*<sup>v</sup>*, *<sup>w</sup>*1), *d*(*<sup>v</sup>*, *<sup>w</sup>*2), ... , *d*(*<sup>v</sup>*, *wk*)). In this sense, the subset of vertices *S* is called a *metric generator* for the graph *G*, if any two distinct vertices produce distinct metric vectors relative to such set *S*. A metric generator of *G* having the minimum possible cardinality is called a *metric basis*, and its cardinality is precisely the *metric dimension* of *G*, which is usually denoted by dim(*G*). The definitions of these concepts (for general metric spaces) are coming from the earliest 1950s from the work [3], although its popularity was not developed until relatively recently (about 15 years before). On the other hand, for the specific case of graphs, and motivated by a problem of uniquely recognizing intruder's locations in networks, these concepts were presented and studied by Slater in [4]. In such work, metric generators were called *locating sets*. On the other hand, Harary and Melter (see [5]) also independently came out with the same concept. In such work, metric generators were called *resolving sets*. It is interesting to remark that some examples of applications of the metric dimension concern navigation of robots in networks as discussed in the work [6], or to chemistry as appearing in [7–9].

An interesting variant of metric dimension in graphs was described by Sebö and Tannier in [10], where they have asked the following question. "*For a given metric generator T of a graph H, whenever H is a subgraph of a graph G, and the metric vectors of the vertices of H relative to T agree in both H and G, is H an isometric subgraph of G?*" The situation is that, despite the fact that metric vectors of all vertices of a graph *G* (relative to a given metric generator) distinguish all pairs of vertices in such graph, it happens that they do not always uniquely recognize all distances in this graph, a fact that was already shown in [10]. Addressed to give a positive answer to their own question, the authors of [10] replaced the notion of "metric generator" by a stronger one. This is described next.

Given a pair of vertices *u*, *v* ∈ *<sup>V</sup>*(*G*), the *interval IG*[*<sup>u</sup>*, *v*] between such two vertices *u* and *v* is defined as the collection of all vertices that belong to some shortest *u* − *v* path. In this sense, a vertex *w strongly resolves* two other different vertices *u* and *v*, if it is satisfied that *v* ∈ *IG*[*<sup>u</sup>*, *w*] or *u* ∈ *IG*[*<sup>v</sup>*, *<sup>w</sup>*], or equivalently, if *dG*(*<sup>u</sup>*, *w*) = *dG*(*<sup>u</sup>*, *v*) + *dG*(*<sup>v</sup>*, *w*) or *dG*(*<sup>v</sup>*, *w*) = *dG*(*<sup>v</sup>*, *u*) + *dG*(*<sup>u</sup>*, *<sup>w</sup>*). In connection with this, it is also said that *u*, *v* are *strongly resolved* by *w*. From now on, all graphs considered are connected. A set *S* of vertices of *G* is a *strong metric generator* for *G* if any two distinct vertices *x*, *y* of such graph are strongly resolved by some vertex *u* ∈ *S* (it could happen that *u* equals *x* or *y*). Then, the smallest possible cardinality of any set being a strong metric generator for *G* is called the *strong metric dimension* of *G*, and this cardinality is denoted by dim*s*(*G*). In addition, a strong metric generator for *G* whose cardinality is precisely equal to dim*s*(*G*) is called a *strong metric basis* of *G*. It is now readily observed that any strong metric generator of *G* also satisfies the property of being a metric generator for *G*. The computational problem concerning finding the strong metric dimension of a given graph is now relatively well studied, and one can find a rich literature concerning it. For more information on this issue, we suggest, for instance, the articles [11,12], the Ph.D. Thesis [13], the survey [14], and references cited therein.

More recently, an extension of the notion of the strong metric dimension of graphs to families of graphs was presented in [15]. The following was stated: Consider that G = {*<sup>G</sup>*1, *G*2, ..., *Gk*} is a family of connected graphs *Gi* = ( *V*, *Ei*) having a common vertex set *V*. Note that the edge sets of the graphs belonging to the family are not necessarily edge-disjoint, and also that the union of their edge sets is not necessarily the complete graph. Concerning such family, it was said in [15] that a *simultaneous strong metric generator* (SSMG for short) for the family G is taken as a set *S* ⊂ *V* with the property that *S* forms a strong metric generator for every graph *Gi* of the family. As usual, an SSMG

having the minimum possible cardinality for G is called a *simultaneous strong metric basis* of G. This smallest cardinality is then precisely called the *simultaneous strong metric dimension* of G, and this is denoted by Sd*s*(G), or by Sd*s*(*<sup>G</sup>*1, *G*2, ..., *Gt*) when it is necessary to clarify the graphs of the family. It is worthwhile mentioning that such concepts arise from a related version of simultaneity for the standard metric dimension studied in [16,17].

The notion of the simultaneous metric dimension of graphs families (and its strong related version) was first studied in the Ph.D. thesis [18], based on the following problem, which arises in relation with a similar problem for the standard metric dimension. It is assumed that the topology of robots navigation network changes within some amount of possible simple networks, say a set (or family) of graphs F. Nodes of the networks remain the same, but their links could appear or disappear. This setting could require the use of a dynamic network whose links change over the time. In this sense, the problem concerning uniquely identifying the robots (by using the smallest resources) navigating in such a "variable" network can be understood as the problem of determining the minimum cardinality of a set of vertices that is simultaneously a metric generator for each graph belonging to this set F. That is, if a set of vertices *S* gives a solution to this problem, then the position of a robot can be uniquely determined by the distance to the elements of *S*, independently of the graph which is being used in each moment in this dynamic network.

We now present some basic terminology and notation to beused throughout our exposition. Given a vertex *v* of a graph *G*, *NG*(*v*) denotes the *open neighborhood* of *v* in *G*, while the *closed neighborhood* is represented by *NG*[*v*] and it equals *NG*(*v*) ∪ {*v*}. If there is no confusion, we then simply use *<sup>N</sup>*(*v*) or *<sup>N</sup>*[*v*]. Two vertices *x*, *y* ∈ *V*(*G*) are called *twins* if they satisfy *NG*[*x*] = *NG*[*y*] or *NG*(*x*) = *NG*(*y*). Specifically, when *NG*[*x*] = *NG*[*y*], they are known as *true twins*, and similarly whether *NG*(*x*) = *NG*(*y*), they are called *false twins*. Now, if the open neighborhood *<sup>N</sup>*(*v*) of a vertex *v* induces a complete graph, then such *v* is known as an *extreme vertex*. The set of extreme vertices of *G* is denoted by *<sup>σ</sup>*(*G*). The largest possible distance between any two vertices of *G* is denoted by *<sup>D</sup>*(*G*), also called the *diameter* of *G*. In this sense, a graph *G* is called 2-antipodal if, for every vertex *x* ∈ *<sup>V</sup>*(*G*), there is exactly one other vertex *y* ∈ *V*(*G*) satisfying the fact that *dG*(*<sup>x</sup>*, *y*) = *<sup>D</sup>*(*G*). Examples of 2-antipodal graphs are, for instance, even cycles *C*2*k*, and the hypercubes *Qr*. Finally, for a given set *W* ⊂ *<sup>V</sup>*(*G*), by *<sup>W</sup><sup>G</sup>*, we represent the subgraph of *G* induced by *W*. Any other definition used shall be introduced whenever a concept is firstly needed.

Since all the definitions above require the connectedness of the graph in question, throughout the whole exposition, we will consider that our graphs are connected; even so, we will not explicitly mention this fact.

### **2. The Simultaneous Strong Resolving Graph**

In this section, we describe an approach which was first presented in [19], in order to transform the problem of finding the strong metric dimension of a graph to computing the vertex cover number of another related graph. To this end, we need some terminology and notation. A vertex *u* of *G* is said to be *maximally distant* from other *v*, if every vertex *w* ∈ *NG*(*u*) satisfies that *dG*(*<sup>v</sup>*, *w*) ≤ *dG*(*<sup>u</sup>*, *<sup>v</sup>*). For a pair of vertices, *u*, *v*, if it happens that *u* is maximally distant from *v* and *v* is also maximally distant from *u*, then these *u* and *v* are called a pair of *mutually maximally distant vertices* (MMD for short). The set of vertices of *G* that are MMD with at least one other vertex of *G* is denoted by *∂*(*G*). The *strong resolving graph* of *G*, which is denoted by *GSR*, is another graph whose vertex set is *V*(*GSR*) = *<sup>V</sup>*(*G*). In addition, there is an edge between two vertices *u*, *v* in *GSR* if such vertices *u* and *v* are mutually maximally distant in the original graph *G*. Clearly, those vertices which are not MMD with any other vertex of *G* are isolated vertices in *GSR*. The recent work [20] (a kind of survey) contains a number of results concerning characterizations, realizability, and several other properties of the strong resolving graphs of graphs.

Now, by a *vertex cover* set of a graph *G*, we mean a set of vertices *S* of *G* satisfying that every edge of *G* has at least one end vertex in the set *S*. The *vertex cover number* of *G*, which is denoted by *<sup>α</sup>*(*G*), is taken as the smallest possible cardinality of a subset of vertices of *G* being a vertex cover set of *G*. By an *α*(*G*)-set, we represent a vertex cover set of cardinality *<sup>α</sup>*(*G*). In connection with this concept, the authors Oellermann and Peters-Fransen (see [19]) have proved that finding the strong metric dimension of a connected graph *G* is equivalent to finding the vertex cover number of *GSR*, which is the next result.

**Theorem 1** ([19])**.** *For any connected graph G,* dim*s*(*G*) = *<sup>α</sup>*(*GSR*).

There are several different and non trivial families of connected graphs for which the strong resolving graphs can relatively easily be obtained. We next mention some of these cases, mainly based on the fact that we further on shall refer to them. Such following observations have already appeared (in an identical presentation) in other works like, for instance [20].
