Secure Italian domination in graphs

An Italian dominating function (IDF) on a graph G is a function f:V(G)→{0,1,2}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:V(G)\rightarrow \{0,1,2\}$$\end{document} such that for every vertex v with f(v)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(v)=0$$\end{document}, the total weight of f assigned to the neighbours of v is at least two, i.e., ∑u∈NG(v)f(u)≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{u\in N_G(v)}f(u)\ge 2$$\end{document}. For any function f:V(G)→{0,1,2}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:V(G)\rightarrow \{0,1,2\}$$\end{document} and any pair of adjacent vertices with f(v)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(v) = 0$$\end{document} and u with f(u)>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(u) > 0$$\end{document}, the function fu→v\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{u\rightarrow v}$$\end{document} is defined by fu→v(v)=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{u\rightarrow v}(v)=1$$\end{document}, fu→v(u)=f(u)-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{u\rightarrow v}(u)=f(u)-1$$\end{document} and fu→v(x)=f(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{u\rightarrow v}(x)=f(x)$$\end{document} whenever x∈V(G)\{u,v}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in V(G){\setminus }\{u,v\}$$\end{document}. A secure Italian dominating function on a graph G is defined as an IDF f which satisfies that for every vertex v with f(v)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(v)=0$$\end{document}, there exists a neighbour u with f(u)>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(u)>0$$\end{document} such that fu→v\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{u\rightarrow v}$$\end{document} is an IDF. The weight of f is ω(f)=∑v∈V(G)f(v)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega (f)=\sum _{v\in V(G) }f(v)$$\end{document}. The minimum weight among all secure Italian dominating functions on G is the secure Italian domination number of G. This paper is devoted to initiating the study of the secure Italian domination number of a graph. In particular, we prove that the problem of finding this parameter is NP-hard and we obtain general bounds on it. Moreover, for certain classes of graphs, we obtain closed formulas for this novel parameter.


Introduction
The following approach to protection of a graph was described by Cockayne et al. (2005). Suppose that one or more guards are stationed at some of the vertices of a simple graph G and that a guard at a vertex can deal with a problem at any vertex in its closed neighbourhood. Consider a function f : V (G) −→ {0, 1, 2} where f (v) is the number of guards at v, and let V i = {v ∈ V (G) : f (v) = i} for every i ∈ {0, 1, 2}. We will identify a function f with the subsets V 0 , V 1 , V 2 of V (G) associated with it, and so we will use the unified notation f (V 0 , V 1 , V 2 ) for the function and these associated subsets. The weight of f is defined to be ω( f ) = f (V (G)) = |V 1 |+2|V 2 |. Informally, we say that G is protected under f if there is at least one guard available to handle a problem at any vertex. Next we show some approaches to the protection of graphs. The functions in each approach protect the graph according to a certain strategy.
We assume that the reader is familiar with the basic concepts, notation and terminology of domination in graphs. If this is not the case, we suggest the textbooks (Haynes et al. 1998a, b). For the remainder of the paper, definitions will be introduced whenever a concept is needed.
A Roman dominating function (RDF) is a function f (V 0 , V 1 , V 2 ) such that for every vertex v ∈ V 0 there exists a vertex u ∈ N G (v) ∩ V 2 , where N G (v) denotes the open neighbourhood of v. The Roman domination number, denoted by γ R (G), is the minimum weight among all RDFs on G. This concept of protection has historical motivation (Stewart 1999) and was formally proposed by Cockayne et al. (2004). A Roman dominating function with minimum weight γ R (G) on G is called a γ R (G)function. A similar agreement will be assumed when referring to optimal functions (and sets) associated with other parameters used in the article.
A generalization of Roman domination, known as Italian domination, was introduced by Chellali et al. (2016) under the name of Roman {2}-domination. The concept was studied further in Henning and Klostermeyer (2017) and Klostermeyer and MacGillivray (2019). An Italian dominating function (IDF) on a graph G is a function The Italian domination number, denoted by γ I (G), is the minimum weight among all IDFs on G.
For any function f (V 0 , V 1 , V 2 ) and any pair of adjacent vertices v ∈ V 0 and u ∈ The weak Roman domination number, denoted by γ r (G), is the minimum weight among all WRDFs on G. This concept of protection was introduced by Henning and Hedetniemi (2003) and studied further in Chellali et al. (2014), Cockayne et al. (2003) and .
Notice that, every γ I (G)-function is a WRDF, which implies that γ I (G) ≥ γ r (G).
A secure dominating set is a dominating set S which satisfies that for every v ∈ V (G)\S there exists u ∈ S ∩ N G (v) such that (S\{u}) ∪ {v} is a dominating set as well. The secure domination number, denoted by γ s (G), is the minimum cardinality among all secure dominating sets. Notice that S is a secure dominating set if and only if there exists a WRDF f (V 0 , V 1 , V 2 ) such that V 1 = S and V 2 = ∅. Hence, γ s (G) ≥ γ r (G). This concept of protection was introduced by Cockayne et al. (2005), and studied further in Merouane and Chellali (2015), Burger et al. (2008), Chellali et al. (2014), Cockayne et al. (2003), Klostermeyer and Mynhardt (2008) and . Now, from the previous inequalities, we derive the following inequality chains.
In this article we introduce the study of secure Italian domination in graphs. We define a secure Italian dominating function (SIDF) to be an IDF f such that the f u→v is an IDF on G. In particular, whenever f u→v is an IDF, we will say that u is a moving neighbour of v. Obviously, if f (V 0 , V 1 , V 2 ) is an SIDF, then every vertex v ∈ V 0 has at least one moving neighbour. The secure Italian domination number, denoted by γ s I (G), is the minimum weight among all SIDFs on G. In Fig. 1   The paper is structured as follows. In Sect. 2 we show that the general problem of finding the secure Italian domination number of a graph is NP-hard. In Sect. 3 we derive general bounds and discuss some extremal cases. In particular, we show that any non-empty graph G satisfies max{γ 2 (G), γ r (G) + 1} ≤ γ s I (G) ≤ γ (G) + γ s (G). In Sect. 4 we obtain a formula for the secure Italian domination number of paths and 2 2 1 2 1 Fig. 1 The labels in these graphs define γ s I (G)-functions cycles. Furthermore, we characterize all trees T satisfying γ s I (T ) = γ (T ) + 1 or γ s I (T ) = γ (T ) + 2. Finally, Sect. 5 is devoted to the study of join graphs.

NP-hardness
This section is devoted to prove that for any graph G of order n there exists a graph G such that γ s I (G ) = γ r (G) + n, showing that the problem of finding γ s I (G ) is as difficult as the problem of finding γ r (G). To begin with, we need to state the following lemma. Let L(G) denotes the set of all leaves in G. Theorem 2 For any graph G of order n, γ s We claim that g is a WRDF on G. We have to show that for every x ∈ W 0 there exists y ∈ W 1 ∪ W 2 such that G does not have undefended vertex under g y→x . We know that for every is an IDF on G • K 2 , and also the restriction of f y→x to V (G) equals g y→x . Thus, y has to belong to W 1 ∪ W 2 and for every z ∈ V 0 \{x} either Our next result shows that we can use rooted product graphs to study the problem of finding the secure Italian domination number of a graph. In this case, the main tool is Theorem 2. It is well known that the weak Roman dominating set problem is an NP-complete decision problem (Henning and Hedetniemi 2003), i.e., given a positive integer k and a graph G, the problem of deciding if G has a weak Roman dominating set D of cardinality |D| ≤ k is NP-complete. Hence, the optimization problem of computing the weak Roman domination number of a graph is NP-hard.

Corollary 3 The problem of computing the secure Italian domination number of a graph is NP-hard.
Proof By Theorem 2, for any graph G of order n we have that γ s I (G • K 2 ) = n + γ r (G). Hence, the problem of computing γ r (G) is equivalent to the problem of finding γ s I (G • K 2 ), which implies that the general problem of computing the secure Italian domination number of a graph is NP-hard.
According to Corollary 3, it would be desirable to obtain tight bound or closed formulas for the secure Italian domination number of a graph. This is precisely the aim of the next sections.

General bounds and extremal cases
To begin this section, we proceed to characterize the graphs achieving the following trivial bounds.

Remark 4 For any graph
Obviously, when characterizing the graphs achieving the bounds we can restrict ourselves to the case of connected non-trivial graphs, and from that characterization it is easy to deduce the result for non-connected graphs.
Theorem 5 Let G be a connected non-trivial graph of order n. Then the following statements hold.
In the first case, neither f x→u nor f y→u is an IDF on G, which is a contradiction. In the second case, we have that f (x) = 0 and neither f u→x nor f v→x is an IDF on G, which is a contradiction again. Therefore, (i) follows.
If G ∼ = K 1,n−1 , then it is easy to observe that γ s I (G) = n. Conversely, suppose γ s I (G) = n and there are two vertices, u, v such that d G (u) ≥ 2 and d G (v) ≥ 2; let us choose u, v such that uv ∈ E(G). We can define an SIDF g on G such that g(u) = 0 and g(x) = 1 for any x ∈ V (G)\{u}. Notice that ω(g) = n − 1 < n = γ s I (G), which is a contradiction. Therefore, (ii) follows.
For any vertex x ∈ V (G) and any γ s I (G)-function f (V 0 , V 1 , V 2 ) such that f (x) = 2, we define the following set associated with x and f , The subgraph of G induced by a set S ⊆ V (G) will be denoted by S . From the definition of SIDF on G we have the following straightforward observation.

Observation 6
If there exists a vertex x ∈ V (G) and a γ s The k-domination number of a graph G, denoted by γ k (G), is the cardinality of a smallest set of vertices such that every vertex not in the set is adjacent to at least k vertices in the set. Such sets are called k-dominating sets. Since every 2-dominating set S is a secure dominating set, and the function f (V 0 , V 1 = S, V 2 = ∅) is an IDF, we conclude that From the inequality γ 2 (G) ≥ γ s (G) and the lower bound stated by the next theorem, we will be able to deduce that the expected inequality γ s I (G) ≥ γ s (G) holds for any graph G.
Theorem 7 For any non-empty graph G, Proof Let D be a γ (G)-set and D a γ s (G)-set. Let us construct a function f from D and D such that We claim that f is an SIDF on G. Obviously, f is an IDF on G. For every vertex . We consider two cases. Case Case 2 x / ∈ N G (x). In this case, there exists y ∈ (D \{y}) ∩ N G (x ) and z ∈ N G (x )∩D. If y = z, then f y→x (y ) = 2, otherwise f y→x (y ) ≥ 1 and f y→x (z) ≥ 1.
According to the two cases above we can conclude that f y→x is an IDF on G. Therefore, f is an SIDF, and so γ s I (G) ≤ ω( f ) = |D| + |D | = γ (G) + γ s (G). Now we prove the lower bound γ s I (G) ≥ γ 2 (G). Let g(V 0 , V 1 , V 2 ) be a γ s I (G)function. Notice that if V 2 = ∅, then V 1 is a 2-dominating set and so γ 2 (G) ≤ |V 1 | = γ s I (G). Assume that V 2 = ∅. Let x ∈ V 2 and notice that if P g (x) = ∅, then the subgraph induced by P g (x) is a clique. With this fact in mind, for every x ∈ V 2 such that P g (x) = ∅ we fix one vertex x ∈ P g (x), and let V g be the set of these representatives. Now, if u ∈ V 0 and u does not belong to any P g (x), then |N G (u) ∩ (V 1 ∪ V 2 )| ≥ 2, while if u ∈ P g (x) for some x ∈ V 2 , then x, x ∈ N G [u]. Thus S = V 1 ∪ V 2 ∪ V g is a 2-dominating set of G. Therefore, γ 2 (G) ≤ |S| = |V 1 |+|V 2 |+|V g | ≤ |V 1 |+2|V 2 | = γ s I (G) and the lower bound γ s I (G) ≥ γ 2 (G) follows. Finally, we proceed to prove the lower bound γ s I (G) ≥ γ r (G) + 1. In this case, let h(W 0 , W 1 , W 2 ) be a γ s I (G)-function. If γ s I (G) = |V (G)|, then we are done, as γ r (G) ≤ |V (G)| − 1 for every non-empty graph. Hence, we fix x ∈ W 0 and y ∈ W 1 ∪ W 2 such that y is a moving neighbour of x. We now construct a weak Roman dominating function h (W 0 , W 1 , W 2 ), which is defined in two different ways depending on whether y ∈ W 2 or not.
Case 1 y ∈ W 2 . In this case, we define the function h by h (y) = 1 and h (z) = h(z) for every z ∈ V (G)\{y}. Since h is an IDF, G does not have undefended vertices under h , i.e., W 1 ∪ W 2 = W 1 ∪ W 2 is a dominating set. Moreover, since for every v ∈ W 0 = W 0 , there exits u ∈ W 1 ∪ W 2 such that h u→v is an IDF, we conclude that G does not have undefended vertices under h u→v . Therefore, h is a weak Roman dominating function on G.
Case 2 y ∈ W 1 . In this case, we define the function h by W 0 = W 0 ∪ {y}, W 1 = W 1 \{y} and W 2 = W 2 . Since h and h y→x are IDFs, we can conclude that G does not have undefended vertices under h . Moreover, for every v ∈ W 0 , there exits u ∈ W 1 ∪ W 2 such that h u→v is an IDF. Hence, if u = y, then G does not have undefended vertices under h u→v . Suppose that y is the only moving neighbour of v ∈ W 0 under h and there exists w ∈ W 0 which is undefended under the function Thus, h y→v (N G (w)) = h(x ) = 1, which is a contradiction. Finally, it is readily seen that the set X = N G (y)∩(W 1 ∪ W 2 ) is not empty and G does not have undefended vertices under the function h z→y for every z ∈ X . Therefore, h is a weak Roman dominating function on G.
Next we show that the bounds above are tight.
• If G is the graph shown in Fig. 2 or the graph shown in Fig. 1 on the right, then γ s Fig. 1 on the left, then γ s . In summary, we can state the following domination chains.

Remark 8 For any non-empty graph G,
For any set S ⊆ V (G) and any pair of different vertices x, y ∈ S, we define the following set.
Theorem 9 If for every γ 2 (G)-set S there exist two vertices x, y ∈ S such that the set S xy contains two non-adjacent vertices, then Proof We claim that if γ s I (G) = γ 2 (G), then there exists a γ 2 (G)-set S such that for every x, y ∈ S either S xy = ∅ or S xy is a clique.
Let g(V 0 , V 1 , V 2 ) be a γ s I (G)-function and S the 2-dominating set constructed in the proof of Theorem 7, i.e., if V 2 = ∅, then S = V 1 , while if V 2 = ∅, then S = V 1 ∪ V 2 ∪ V g . Since γ s I (G) = γ 2 (G), we have that S is a γ 2 (G)-set. Suppose to the contrary that there exist two different vertices x, y ∈ S and two different nonadjacent vertices u, v ∈ V 0 such that u, v ∈ S x,y . We differentiate the following cases.
Case 1 x, y ∈ V 1 . Since u and v are not adjacent, neither g x→u nor g y→u is an IDF, which is a contradiction, as g is a γ s I (G)-function. Case 2 x ∈ V 1 and y ∈ V 2 . Since S xy is not a clique and g is a γ s I (G)-function, P g (y) = ∅. Hence, |V g | < |V 2 |, which implies that γ 2 (G) = |S| = |V 1 |+|V 2 |+|V g | < |V 1 | + 2|V 2 | = γ s I (G), and this is a contradiction. Case 3 x, y ∈ V 2 . Since S xy = ∅ and g is a γ s I (G)-function, P g (x) = ∅ or P g (y) = ∅. Hence, |V g | < |V 2 | and, as in Case 2, we arrive to a contradiction.
Case 4 x ∈ V g . If x ∈ P g (y), then S xy ⊆ P g (y) and P g (y) is a clique, which is a contradiction. Now, if y ∈ V g \{x}, then S xy = ∅, which is a contradiction. Finally, if y ∈ V 1 ∪ V 2 and x / ∈ P g (y), then u, v ∈ P g (y), which is a contradiction, as P g (y) is a clique.
According to the four cases above, the proof of our claim is complete. Therefore, if for every γ 2 (G)-set S there exist two vertices x, y ∈ S such that the set S xy contains two non-adjacent vertices, then γ s I (G) = γ 2 (G), and by Theorem 7 we conclude that γ s I (G) ≥ γ 2 (G) + 1. The bound above is tight. To see this we can take, for instance, the graph shown in Fig. 1, on the right. Now, in order to show that the converse of Theorem 9 does not hold, we can consider the graph shown in Fig. 1, on the left, where γ s I (G) = 5 = γ 2 (G) + 1 and the γ 2 (G)-set S consisting of all vertices of degree five satisfies that S xy is a clique for every x, y ∈ S. Another example in the same direction is the graph sown in Fig. 2, where γ s I (G) = 7 = γ 2 (G) + 1 and the γ 2 (G)-set S consisting of all black-coloured vertices, except the central one, satisfies S xy = ∅ for every x, y ∈ S.

The particular case of cycles and trees
Next we obtain closed formulas for the secure Italian domination number of cycles and paths. For this purpose, we shall need the following lemma.
Lemma 10 If G ∼ = P n or G ∼ = C n , then there exists a γ s Proof It readily seen that the result follows for n ≤ 5. Assume that n ≥ 6. Let For n ≡ i (mod 5) we define l = n−i 5 and consider a partition l of V (C n ) defined as follows. If i = 0, then l = {X 0 , . . . , X l−1 } and if i ≥ 1, then l = {X 0 , . . . , X l−1 , X l }. For any j < l, the set X j = {v 5 j , v 5 j+1 , . . . , v 5 j+4 } contains five consecutive vertices of C n , while X l contains the remaining i consecutive vertices. For We can see these weights as l consecutive sequences of numbers 10110. The weight of the vertices in X l is assigned as follows.
The weight of f can be expressed in terms of n ≡ i (mod 5) as follows.
, then x belongs to a sequence of consecutive vertices axbc with sequence of weights 1011, or belongs to a sequence of consecutive vertices abxc with sequence of weights 1101. In both cases, f b→x is an IDF. Hence, f is an SIDF and so γ s be a γ s I (C n )-function which satisfies Lemma 10. Since V 2 = ∅, we have that V 1 is a 2-dominating set, and so for Thus, for every sequence of five consecutive vertices, at most two of them can belong to V 0 , i.e., (1) Hence, Therefore, γ s I (C n ) ≥ 3n 5 . Let G be a graph and e ∈ E(G). The edge-deletion subgraph G − e of G is defined to be G − e = (V (G), E(G)\{e}). Observe that every γ s I (G − e)-function is an SIDF on G, which implies the following result.

Theorem 12 For any spanning subgraph H of a graph G,
From Proposition 11 and Theorem 12 we derive the following consequence.
Theorem 13 For any Hamiltonian graph G of order n, γ s I (G) ≤ 3n 5 .
As an example of graph G C n where the bound above is achieved, we take the graph shown in Fig. 1, on the left.

Proposition 14
For any non-trivial path P n , Proof The cases where n ≤ 4 are easy to check. Hence, from now on we assume that n ≥ 5. Let f (V 0 , V 1 , V 2 ) be a γ s I (P n )-function which satisfies Lemma 10. In this proof we adapt the procedure developed in the proof of Proposition 11 to the case of paths. Thus, we assume that V (P n ) = {v 0 , v 1 , . . . , v n−1 }, where consecutive vertices are adjacent. From Eq. (1) we know that ψ i ≥ 3 for every i ∈ {0, . . . , n}. In this case, the weight of the leaves has to be one, i.e., f has to be 1011 or 1101. Analogously, the sequence f (v n−4 ) f (v n−3 ) f (v n−2 ) f (v n−1 ) has to be 1101 or 1011. In all cases, which implies that γ s I (P n ) ≥ 3n+2 5 . To conclude the proof we only need to construct an SIDF as described in the proof of Proposition 11, with the only difference that, for n ≡ 0 (mod 5) we take f (v n−1 ) = 1 and for n ≡ 3 (mod 5) we take f (v n−2 ) = 1. In this case, the weight of f can be expressed as follows.
From Remark 8 (c) we learned that γ s I (G) ≥ γ (G) + 1 for every non-empty graph G. We proceed to characterize the trees T satisfying the equalities γ s I (T ) = γ (T ) + 1 or γ s I (T ) = γ (T ) + 2. We begin with the following observation. Observation 15 If T is a tree of order n ≥ 3, then always is possible to choose a γ (T )-set not containing any leaf of T .
From Observation 15 and Lemma 1 we have the following lemma.
Lemma 16 Let T be a tree. If γ s I (T ) = γ (T ) + 1, there there exists at most one strong support s in T and s is the neighbour of exactly two leaves.
From the result above we deduce the following corollary.