*3.1. Unicyclic Graphs*

Given a unicyclic graph *G* different from a cycle, from now on we will denote by *Cr* = *v*1*v*2 ... *vrv*1 the subgraph induced by the unique cycle of *G*. A vertex *v* ∈ *V*(*G*) of degree one is a *terminal vertex* of *G*, and *T*(*G*) is the set of terminal vertices of *G*. Note that the terminal vertices defined here represent a particular case of the terminal vertices defined for cactus graphs in general. If the vertex *vi* of *Cr* has degree greater than two, then we say that *ui* is a terminal vertex of *vi*, if *dG*(*ui*, *vi*) = min{*dG*(*ui*, *vj*,) : *vj* = *vi*}. The set of terminal vertices of a vertex *vi* is denoted by *<sup>t</sup>*(*vi*). We will denote by *<sup>c</sup>*2(*G*) the set of vertices of the cycle *Cr* having degree two. If *v* ∈ *<sup>c</sup>*2(*G*), then we will say that *<sup>t</sup>*(*v*) = ∅.

Notice that if the unicyclic graph *G* is isomorphic to the cycle *Cn*, then for *n* even (*Cn*)*SR* ∼= *n* 2 *i*=1 *K*2 and for *n* odd (*Cn*)*SR* ∼= *Cn* as already presented in Observation 2. Thus, we will study the cases that *G* ∼= *Cn*.

We begin with the following straightforward observations that are useful to describe the strong resolving graph of any unicyclic graph.

**Remark 4.** *Let G be a unicyclic graph. For every vertex x* ∈ *<sup>c</sup>*2(*G*) *there exists at least one vertex y* ∈ *<sup>c</sup>*2(*G*) ∪ *T*(*G*) *such that x*, *y are MMD in G.*

**Remark 5.** *Let G be a unicyclic graph. Then two vertices x*, *y are MMD in G if and only if x*, *y* ∈ *<sup>c</sup>*2(*G*) ∪ *<sup>T</sup>*(*G*)*.*

**Corollary 3.** *For any unicyclic graph G, ∂*(*G*) = *<sup>c</sup>*2(*G*) ∪ *<sup>T</sup>*(*G*)*.*

Notice that every two vertices *x*, *y* ∈ *T*(*G*) are MMD. Also, every vertex *v* ∈ *<sup>c</sup>*2(*G*) is MMD with every vertex *w* satisfying one of the following conditions.


As a consequence of the above comments, we can deduce the structure of the strong resolving graph of any unicyclic graph *G* in the following way. First notice that, according to Corollary 3, *GSR* has vertex set equal to *<sup>c</sup>*2(*G*) ∪ *<sup>T</sup>*(*G*), and to describe the adjacency of vertices in *GSR* we consider two cases.

*GSR* for *r* even.


• If *x*, *y* are diametral vertices in *Cr*, *x* ∈ *<sup>c</sup>*2(*G*) and *y* ∈/ *<sup>c</sup>*2(*G*), then {*x*} ∪ *<sup>t</sup>*(*y*) forms a subgraph of *GSR* isomorphic to *<sup>K</sup>*|*t*(*y*)|+<sup>1</sup> and *NGSR* (*x*) = *<sup>t</sup>*(*y*).

As a consequence of the description above, we can observe that *β*(*GSR*) ≤ |*<sup>c</sup>*2(*G*)|−<sup>1</sup> 2+ |*T*(*G*)|.

*GSR* for *r* odd.

	- **–** If *x*, *y* ∈ *<sup>c</sup>*2(*G*), then {*<sup>u</sup>*, *x*, *y*} is a subgraph of *GSR* isomorphic to *P*3, *NGSR* (*u*) = {*<sup>x</sup>*, *y*} and for every *w* ∈ {*<sup>x</sup>*, *y*}, *<sup>δ</sup>GSR* (*w*) ≥ 2.
	- **–** If *x*, *y* ∈/ *<sup>c</sup>*2(*G*), then {*u*} ∪ *<sup>t</sup>*(*x*) ∪ *<sup>t</sup>*(*y*) is a subgraph of *GSR* isomorphic to *<sup>K</sup>*|*t*(*x*)|+|*t*(*y*)|+1, *NGSR* (*u*) = *<sup>t</sup>*(*x*) ∪ *<sup>t</sup>*(*y*) and for every *w* ∈ *<sup>t</sup>*(*x*) ∪ *<sup>t</sup>*(*y*), *<sup>δ</sup>GSR* (*w*) ≥ |*t*(*x*)| + |*t*(*y*)| + 1 for *r* ≥ 5 (notice that if *r* = 3, then *<sup>δ</sup>GSR* (*w*) = |*t*(*x*)| + |*t*(*y*)|).
	- **–** If *x* ∈ *<sup>c</sup>*2(*G*) and *y* ∈/ *<sup>c</sup>*2(*G*), then the set {*<sup>u</sup>*, *x*} ∪ *<sup>t</sup>*(*y*) form a subgraph (not induced) (Notice that the vertices *<sup>t</sup>*(*y*) are adjacent between them in *GSR*.) of *GSR* isomorphic to a star graph *<sup>S</sup>*1,|*t*(*y*)|+<sup>1</sup> with central vertex *u*, *NGSR* (*u*) = {*x*} ∪ *<sup>t</sup>*(*y*), *<sup>δ</sup>GSR* (*x*) ≥ 2 and for every *w* ∈ *<sup>t</sup>*(*y*), *<sup>δ</sup>GSR* (*w*) ≥ |*t*(*y*)| + 1.

Similarly to the case when *r* is even, we can observe here that *β*(*GSR*) ≤ |*<sup>c</sup>*2(*G*)| 2 + |*T*(*G*)|.

We define the *branch restricted unicyclic graph* T (*G*) associated to a unicyclic graph *G* in the following way. We begin with taking the cycle *Cr* in *G* and removing the remaining vertices of *G*. Then we add *<sup>t</sup>*(*vi*) pendant edges to every vertex *vi* in *Cr*. Figure 1 shows an example of a unicyclic graph, its branch restricted unicyclic graph and its strong resolving graph.

**Figure 1.** A unicyclic graph *G*, T (*G*) and *GSR*.

**Lemma 1.** *Let G be a unicyclic graph and* T (*G*) *be its branch restricted unicyclic graph. Then* (T (*G*))*SR is isomorphic to GSR*

**Proof.** From Remarks 4 and 5, and by the definition of the branch restricted unicyclic graph, we deduce that (T (*G*))*SR* is isomorphic to *GSR*.

Our next step is dedicated to present a realization result for some corona product graphs, where the solution precisely involves the use of unicyclic graphs. We first recall that the *corona product graph G H* is defined as the graph obtained from a graph *G* of order *n* and a graph *H*, by taking one copy of *G* and *n* copies of *H*, and then joining by an edge each vertex from the *<sup>i</sup>th*-copy of *H* with the *ith*-vertex of *G*.

**Proposition 1.** *For any integer n* ≥ 3*, there exists a graph G such that GSR* ∼= *Kn K*1*.*

**Proof.** We consider the unicyclic graph *G* with a cycle *C*2*n* = *v*1*v*2 ... *v*2*nv*1 such that the vertices *v*1, *v*2,..., *vn* form the set *<sup>c</sup>*2(*G*) and the remaining ones from the cycle have exactly one terminal vertex. Since 2*n* is an even number according to the Description of *GSR* it clearly follows that *GSR* is isomorphic to *<sup>K</sup>*|*T*(*G*)<sup>|</sup> where each vertex of *T*(*G*) has exactly one neighbor in *<sup>c</sup>*2(*G*).

### *3.2. Bouquet of Cycles*

Let B*<sup>a</sup>*,*b*,*<sup>c</sup>* be a family of graphs obtained in the following way. Each graph *B* ∈ B*<sup>a</sup>*,*b*,*<sup>c</sup>* is a bouquet of *a* + *b* + *c* cycles where *a* of them are even cycles (of order at least four), *b* are odd cycles of order larger than three, *c* are cycles of order three, *a*, *b*, *c* ≥ 0, and *a* + *b* + *c* ≥ 2. All cycles of *B* ∈ B*<sup>a</sup>*,*b*,*<sup>c</sup>* have the common vertex *w*. One example of a bouquet of cycles is given in Figure 2. Let *Cr*1 , *Cr*2 , ... , *Cra* be the even cycles of order at least four in *B* ∈ B*<sup>a</sup>*,*b*,*<sup>c</sup>* and *Cs*1 , *Cs*2 , ... , *Csb* be the odd cycles of order larger than three in *B* ∈ B*<sup>a</sup>*,*b*,*c*.

**Figure 2.** A bouquet of cycles *B* ∈ B2,2,1 containing the cycles *C*6, *C*4, *C*9, *C*7 and *C*3.

In [17], the authors have described the structure of the strong resolving graph of the graph *B* ∈ B*<sup>a</sup>*,*b*,*<sup>c</sup>* as follows. By completeness of our exposition, we copy exactly the description presented there, since it makes no sense to do some changes on it, as it is fairly well written.


Figure 3 shows the strong resolving graph of the graph illustrated in Figure 2.

**Figure 3.** The strong resolving graph *BSR* of the graph illustrated in Figure 2.

If we study the bouquet of cycles *B* ∈ B*<sup>a</sup>*,*b*,*<sup>c</sup>* with *b* = 0 (or equivalently, *B* has not odd cycles of order larger than three), and *Cr*1 , *Cr*2 , ... , *Cra* are the cycles of even order, then the strong resolving graph *BSR* is composed by the complete graph *Ka*+2*c* and ∑*ai*=<sup>1</sup> *ri*−2 2 components isomorphic to *K*2.

Now, we again give some realization results for strong resolving graphs. To this end, we need to define a graph structure which we call a *partial multisubdivided complete graph <sup>K</sup>*2*n*(*p*1, *p*2, ... , *pn*). That is, a complete graph *K*2*n* where each edge of a perfect matching of this graph is subdivided *pi* ≥ 0 times for *i* ∈ {1, 2, ..., *n*} (the case when some *pi* = 0 means that the edge corresponding to *pi* is not subdivided). Moreover, recall that the *cocktail party graph Rn*, also called the hyperoctahedral graph, is a *n* − 2 regular graph on *n* vertices.

**Proposition 2.** *For any integer n* ≥ 2*, there exists a graph G such that GSR is isomorphic to <sup>K</sup>*2*n*(*p*1, *p*2,..., *pn*)*.*

**Proof.** We consider the bouquet of cycles *B* ∈ B*<sup>a</sup>*,*b*,*<sup>c</sup>* with *a*, *c* = 0, *b* = *n* and *Cp*1+3, *Cp*2+3, ... , *Cpn*+<sup>3</sup> are the cycles of odd order larger than three. According to the construction of the strong resolving graph *BSR*, the subgraph *Vb* is isomorphic to *R*2*n* and the set of vertices of each odd cycle *Cpi*+3, *i* ∈ {1, ... , *b*}, which are different from *w* induces a path of order *pi* + 2, in *BSR*, whose leaves are the two vertices of this cycle that are not adjacent in *R*2*n*.

**Corollary 4.** *For any integer n* ≥ 2*, there exists a graph G such that GSR contains the cocktail party graph R*2*n as an induced subgraph.*

### *3.3. Chains of Even Cycles*

A *chain of cycles* is a cactus graph in which, every cycle has order at least three and there are only two terminal cycles. Notice that in such case every non-terminal cycle has exactly two cut vertices, such that each cut vertex belongs to exactly two cycles. We next center our attention into the case of chains of even cycles. To this end, we need some terminology and notation. A chain of even cycles is a *straight chain*, if the cut vertices of every cycle in the chain are diametral in the cycle. Note that each straight chain contains two diametral vertices, which are the unique terminal vertices of this chain.

For the purposes of simplifying, given an integer *k* ≥ 0, we shall define the next family F*k* of graphs. Each graph *F* ∈ F*k* is a chain of even cycles constructed as follows.


Notice that for instance, for the chain of even cycles *F* ∈ F*k* described above, the two terminal vertices of it are *a*0 and *b*0. Figure 4 shows a fairly representative example of a chain of even cycles. Recall that the way of drawing such graph (with respect to directions of the "turns" in the chain) does not influence in our purposes. The chain of even cycles *F* ∈ F*k* presented in the Figure 4 has four straight chains of even cycles: *G*0 contains *C*<sup>1</sup> and *C*2, *G*1 contains *C*2, *C*<sup>3</sup> and *C*4, *G*2 contains *C*4, *C*<sup>5</sup> and *C*6, and *G*3 contains *C*6, *C*<sup>7</sup> and *C*<sup>8</sup> .

**Figure 4.** A chain of cycles *F* ∈ F3 containing six cycles *C*4 and two cycles *C*6.

We next describe the strong resolving graph of a chain of even cycles *F* ∈ F*k*. We need first the following observations.

**Remark 6.** *For any chain of even cycles F* ∈ F*k, a vertex x belongs to ∂*(*F*) *if and only if x has degree two.*

**Remark 7.** *In a straight chain of cycles, the two terminal vertices form a pair of MMD vertices, as well as each pair of diametral vertices in each cycle.*

**Observation 3.** *For a chain of even cycles F* ∈ F*k, and for every i* ∈ {0, . . . , *k*} *and j* ∈ {*<sup>i</sup>*,..., *k*} *it follows.*


For instance, in Figure 4, the red vertex *a*1 is MMD with the blue vertices *b*2, *b*1, *b*0, while the blue vertex *b*1 is MMD with the red vertices *a*2, *a*1, *a*0. Moreover, again in Figure 4, any pair of green diametral vertices belonging to the same cycle are MMD in *F*.

With these observations above, we are able to describe the structure of *FSR* for every chain of even cycles *F* ∈ F*k*. To do so, we shall need the following construction, which represents a bipartite graph *Jr* of order 2*r* + 2 for some *r* ≥ 3. The two bipartition sets of the bipartite graph *Jr* are the sets *U* = {*<sup>a</sup>*0, ... , *ar*} and *V* = {*b*0, ... , *br*}. The edges of *Jr* are as follows. For every *i* ∈ {0, ... , *r*/2} and every *j* ∈ {0, . . . ,*r* − *i*}, there exist the edges *aibj* and *biaj*.


We may remark that, the strong resolving graph of a straight chain of cycles is simply a union of several complete graphs *K*2. The strong resolving graph of the chain of even cycles shown in Figure 4 is drawn in Figure 5.

**Figure 5.** The strong resolving graph *FSR* of the graph illustrated in Figure 4.

We end this subsection by giving a realization result for strong resolving graphs involving chains of even cycles.

**Corollary 5.** *For any integer k* ≥ 2*, there exists a chain of even cycles F* ∈ F*k such that FSR contains the bipartite graph Jk as a component.*

### **4. The Strong Metric Dimension**

We are next centered into computing or bounding the strong metric dimension of the cactus graphs which we have studied in the previous section.
