*Notation*

Throughout this article we consider *G* = (*V*(*G*), *E*(*G*)) as a simple graph of order *n* = |*V*(*G*)|. Given a vertex *v* of *G*, *<sup>N</sup>*(*v*) and *<sup>N</sup>*[*v*] represent the open neighbourhood and the closed neighbourhood of *v*, respectively. For a set *D* ⊆ *<sup>V</sup>*(*G*), its open neighbourhood and closed neighbourhood are *N*(*D*) = <sup>∪</sup>*v*∈*D<sup>N</sup>*(*v*) and *N*[*D*] = *N*(*D*) ∪ *D*, respectively. The boundary of the set *D* is defined as *∂*(*D*) = *N*(*D*) \ *D*. The private neighbourhood of a vertex *v* with respect to a set *D* ⊆ *V*(*G*) (*v* ∈ *D*), denoted by *pn*(*<sup>v</sup>*, *<sup>D</sup>*), is defined by *pn*(*<sup>v</sup>*, *D*) = {*u* ∈ *<sup>V</sup>*(*G*): *<sup>N</sup>*(*u*) ∩ *D* = {*v*}}. The vertices of *pn*(*<sup>v</sup>*, *D*) will be called private neighbours of *v* with respect to *D*. Given a vertex *v* ∈ *D* ⊆ *<sup>V</sup>*(*G*), *epn*(*<sup>v</sup>*, *D*) = *pn*(*<sup>v</sup>*, *D*) ∩ (*V*(*G*) \ *D*) represent the external private neighbourhood of *v* with respect to *D*. Also, and as is commonly defined, *G* − *D* denotes the graph obtained from *G* such that *V*(*G* − *D*) = *V*(*G*) \ *D* and *E*(*G* − *D*) = *E*(*G*) \ {*uv* ∈ *E*(*G*) : *u* ∈ *D* or *v* ∈ *<sup>D</sup>*}. The subgraph induced by *D* ⊆ *V*(*G*) is denoted by *<sup>G</sup>*[*D*]. For any two vertices *u* and *v*, the distance *d*(*<sup>u</sup>*, *v*) between *u* and *v* is the length of a shortest *u* − *v* path.

A set *X* of vertices of *G* is a packing in *G* if the closed neighbourhoods of vertices in *X* are pairwise disjoint, that is, if *<sup>N</sup>*[*u*] ∩ *<sup>N</sup>*[*v*] = ∅, for every pair of different vertices *u*, *v* ∈ *X*.

A leaf vertex of a graph *G* is a vertex of degree one, and a support vertex of *G* is a vertex adjacent to a leaf. The set of leaves and support vertices are denoted by *L*(*G*) and *<sup>S</sup>*(*G*), respectively. Also, given a set *D* ⊆ *V*(*G*) we denote *I*(*D*) as an independent set of maximum cardinality in *G*[*D*] such that |*I*(*D*) ∩ *S*(*G*)| is maximum.

Other definitions will be introduced as needed.
