Saturated boundary k -alliances in graphs

Discrete Applied Mathematics 185 (2015) 192–207

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Saturated boundary k-alliances in graphs Hachem Slimani a,∗ , Hamamache Kheddouci b a

LaMOS Research Unit, Computer Science Department, University of Bejaia, 06000 Bejaia, Algeria

b

Université de Lyon, CNRS, Université Lyon 1, LIRIS, UMR5205 F-69622, France

article

info

Article history: Received 7 December 2013 Received in revised form 13 November 2014 Accepted 26 November 2014 Available online 22 December 2014 Keywords: Saturated vertex Saturated boundary k-alliance Boundary defensive (−1)-alliance Boundary offensive 1-alliance Boundary powerful (−1)-alliance Minimal (connected) global boundary powerful (−1)-alliance

abstract In this paper, we introduce a new concept of saturated vertices for the alliances. For a given graph G = (V , E ) and S ⊂ V , a vertex v ∈ V is said to be S-saturated, if the number of its defenders is equal to the number of its attackers for S. We define new parameters |λ(S )| and ¯ S )| as the number of S-saturated vertices and not S-saturated vertices of S, respectively. |λ( We study mathematical properties of (global) saturated boundary defensive, offensive and powerful k-alliances and theoretical results are obtained by giving in particular some tight bounds and exact values on the cardinality of such alliances. As a main result, we give tight bounds for the cardinality of every minimal global boundary powerful (−1)-alliance (MGBPA) in terms only of the order and the size of graph. Furthermore, we establish two algorithms which generate graphs containing (connected) MGBPA from a given complete graph K2n . © 2014 Elsevier B.V. All rights reserved.

1. Introduction The study of alliances in graphs by giving their mathematical properties was initiated by Kristiansen et al. [13] (see also Haynes et al. [11]). They defined alliances of different types that have been extensively studied during the last recent years. These alliance types are called defensive alliances [21,28], offensive alliances [6,14,22] and dual alliances or powerful alliances [1,2]. A generalization of these alliances called k-alliance (or r-alliance) introduced by Shafique and Dutton [19,20] has received more attention in recent years. There have been studies on (global) defensive/offensive k-alliances [10,16,17], boundary defensive/powerful k-alliances [26,27], partitioning a graph into k-alliances [23,25,29], k-alliances with (k)domination in graphs [8,9], exact defensive k-alliances and alliance polynomials [3,4]. Some of these alliances have been studied for certain classes of graphs including, but are not limited to, tree, path, cycle, star, planar and line graphs [5,12, 13,15,21]. For more details and discussion, one can see the surveys by Fernau and Rodríguez-Velázquez [7] and Yero and Rodríguez-Velázquez [28]. In practice, an alliance can be a bond or connection between individuals, families, states, or parties, etc. In [13], Kristiansen et al. considered alliances of nations at war and in the corresponding graph the vertices represent the nations and the edges correspond to possible relations (of either friendship or hostility) between them. In a defensive alliance every member has more bonds in the alliance (including itself) than outside the alliance. In this case, we say that each member of the alliance has more defenders than attackers. Thus, in a defensive alliance all members are defended from possible attack by non-members. However, in an offensive alliance every non-member has more bonds in the alliance than outside the alliance (including itself). In this case, we say that each non-member of the alliance has more attackers than defenders. Thus, in an offensive alliance all non-members are vulnerable to possible attack by members of the alliance. Note that each (non)-member is a defender of itself. Powerful alliances are both defensive and offensive.

∗

Corresponding author. Tel.: +213 34 21 08 00; fax: +213 34 21 51 88. E-mail addresses: [email protected] (H. Slimani), [email protected] (H. Kheddouci).

http://dx.doi.org/10.1016/j.dam.2014.11.030 0166-218X/© 2014 Elsevier B.V. All rights reserved.

H. Slimani, H. Kheddouci / Discrete Applied Mathematics 185 (2015) 192–207

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Fig. 1. Critical state of the boundary defensive/powerful k-alliances.

In [26,27], Yero and Rodríguez-Velázquez have defined a variant of alliance called boundary defensive (resp. offensive) k-alliance in a graph G = (V , E ) as a set S of vertices of G with the property that every vertex in S (resp. ∂ S) has exactly k more neighbors in S than it has outside of S with k ∈ {−∆, . . . , ∆} (resp. k ∈ {2 − ∆, . . . , ∆}), where ∂ S denotes the set of neighbors of S in V − S and ∆ is the maximum degree. They also have defined (global) boundary powerful k-alliance as a set S of vertices which is both (global) boundary defensive k-alliance and (global) boundary offensive (k + 2)-alliance. Yero and Rodríguez-Velázquez have studied mathematical properties of such alliances by obtaining in particular several bounds on the cardinality of every (global) boundary defensive/powerful k-alliance. Note that every boundary k-alliance S has a limit depending on k such that over this limit S becomes a strong alliance and under this limit S looses the property of alliance. This limit is k = −1 for boundary defensive and powerful k-alliances, and k = 1 for boundary offensive k-alliances. Thus, at this limit (state), which corresponds to the border where the alliance can switch from strong to no alliance, the stability of a boundary k-alliance is fragile and at risk as illustrated in Fig. 1. In this paper, we study alliances which are in this critical state corresponding to alliances where each member (inside or outside the alliance) has as many defenders as attackers (i.e. for each member, the number of its defenders is equal to the number of its attackers). We call the alliances having this property by saturated boundary k-alliances. Thus we have two kinds of alliances: boundary defensive/powerful (−1)-alliances and boundary offensive 1-alliances. It is interesting to consider alliances of this type because they present a very sensitive situation in the sense that adding a new non-member or bonds between non-members (between non-members and members) or removing a member or bonds between members (between members and non-members) will necessarily change the structure of the alliance. In other words, adding (resp. removing) a vertex or an edge to (resp. from) the corresponding graph can disrupt the alliance, which makes its stability fragile and at risk. Thus, we introduce a new concept of S-saturated vertices in a graph G = (V , E ) as follows: for S ⊂ V , a vertex v ∈ V is said to be S-saturated, if the number of its defenders is equal to the number of its attackers for S. We study mathematical properties for saturated boundary defensive, offensive and powerful k-alliances. We obtain theoretical results by defining ¯ S )| which represent, respectively, the number of S-saturated vertices and the and using new parameters |λ(S )| and |λ( ¯ S )| ̸= 0 for every global defensive (−1)-alliance S which is not number of not S-saturated vertices of S. We prove that |λ( minimal. (An alliance S (of some type) is called minimal if no proper subset of S is an alliance (of the same type).) In order to prove our main result, where tight bounds will be given for the cardinality of every minimal global boundary powerful (−1)-alliance (MGBPA), we need to prove several lemmas. Moreover, we establish two algorithms which generate graphs containing (connected) MGBPA from a given complete graph K2n (S ⊂ V is connected (resp. not connected) if its induced subgraph ⟨S ⟩ is connected (resp. not connected)). The paper is structured as follows: in Section 2, we give some preliminaries and definitions that will be useful in the rest of the paper. In Section 3, we introduce the concept of S-saturated vertices and we give some results for (saturated) defensive/offensive k-alliances that will be used in Section 4, where we study the global boundary powerful (−1)-alliances by giving several tight bounds for the parameter |λ(S )|. In Section 5, we define some procedures that will be used in Section 6 to establish two algorithms which generate graphs containing (connected) MGBPA. Finally, Section 7 presents our conclusion. 2. Preliminaries and definitions We define a graph G to be a pair (V , E ) where V denotes the vertex set and E to be the edge set in the graph. For any vertex v ∈ V , we define N (v) to be the neighborhood of the vertex v , i.e. the set of vertices that are adjacent to v in G, and N [v] = N (v) ∪ {v}. dS (v) = |NS (v)| is the degree of v in S with NS (v) = {u ∈ S : uv ∈ E } is the set of neighbors v has in S. Now, we recall the definitions of defensive, offensive and powerful k-alliances. Definition 1 ([19,20]). A non-empty set of vertices S ⊆ V is a defensive k-alliance in G = (V , E ), k ∈ {−∆, . . . , ∆}, if for every v ∈ S: dS (v) ≥ dV −S (v) + k,

(1)

where ∆ is the maximum degree. A defensive (−1)-alliance is a defensive alliance and a defensive 0-alliance is a strong defensive alliance as defined in [13]. A defensive 0-alliance is also known as a cohesive set [18]. In these cases, by strength of numbers, we say that every vertex in S is (strongly) defended from possible attack by vertices in V − S. The alliance S is called global if it effects every vertex in V − S (every vertex in V − S is adjacent to at least one member of the alliance S). In this case, S is a dominating set.

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Fig. 2. Saturated and not saturated vertices in global defensive k-alliances.

Definition 2 ([19,20]). A non-empty set of vertices S ⊆ V is an offensive k-alliance in G = (V , E ), k ∈ {2 − ∆, . . . , ∆}, if for every v ∈ ∂ S = N [S ] − S, the relation (1) is satisfied. In particular, an offensive 1-alliance is an offensive alliance and an offensive 2-alliance is a strong offensive alliance as defined in [13]. In these cases, since each vertex in ∂ S has more neighbors in S than in V − S, we say that every vertex in ∂ S is (strongly) vulnerable to possible attack by vertices in S. The alliance S is called global if the relation (1) is satisfied for every v ∈ V − S (i.e. ∂ S = V − S). Definition 3 ([8,9]). A non-empty set of vertices S ⊆ V is a powerful k-alliance in G = (V , E ), k ∈ {−∆, . . . , ∆ − 2}, if it is both defensive k-alliance and offensive (k + 2)-alliance in G. The case k = −1 corresponds to the standard powerful alliances defined in [1]. Yero and Rodríguez-Velázquez [26,27], by considering the limit case in k-alliances, have defined the so-called boundary defensive (resp. offensive and powerful) k-alliances. Definition 4. Given a graph G = (V , E ),

• a set S ⊆ V is a boundary defensive k-alliance in G, k ∈ {−∆, . . . , ∆}, if dS (v) = dV −S (v) + k, ∀ v ∈ S ; • a set S ⊆ V is a boundary offensive k-alliance in G, k ∈ {2 − ∆, . . . , ∆}, if dS (v) = dV −S (v) + k,

∀ v ∈ ∂ S;

• a set S ⊆ V is a (global) boundary powerful k-alliance in G, k ∈ {−∆, . . . , ∆ − 2}, if S is both (global) boundary defensive k-alliance and (global) boundary offensive (k + 2)-alliance in G. 3. S-saturated vertices in defensive/offensive k-alliances A vertex in a graph is said to be a defender of a neighboring vertex, if both the vertices are in the alliance or both the vertices are not in the alliance. Also, a vertex is considered a defender of itself. A vertex is said to be an attacker of an adjacent vertex, if one of them is in the alliance but the other one is not. A vertex is called defended, if the number of its defenders is greater or equal to the number of its attackers. And a vertex is called to be attacked, if the number of its attackers is greater or equal to the number of its defenders [24, p27]. Let us study the limit case where for every vertex the number of its attackers is equal to the number of its defenders with respect to a given subset S. Thus, we introduce the following concept. Definition 5. Let G = (V , E ) be a graph and S a nonempty subset of V . A vertex v ∈ V is said to be S-saturated, if

|N [v] ∩ S | = |N [v] ∩ (V − S )|. Equivalently, a vertex v ∈ S (resp. v ∈ V − S) is said to be S-saturated, if dS (v) + 1 = dV −S (v) (resp. dS (v) = dV −S (v) + 1). For a graph G = (V , E ) and S ⊂ V , we denote by ES the set of edges of E which have its two extremities in S, i.e. the edges of ⟨S ⟩ which is the subgraph of G induced by S. We also consider ω(S ) the cut or cocycle induced by S, which is the set of edges of E which have an extremity in S and the other in V − S, i.e. the set of all the edges of E crossing the partition {S , V − S } of E. Example 1. Let us consider the graphs (see Fig. 2). For S1 = {a, b} and S2 = {a, b, c }, we have: the vertex b is not S2 -saturated but it is S1 -saturated. However, the vertex a is neither S1 -saturated nor S2 -saturated. Note that S2 is a global defensive 0-alliance but it is not minimal, and all vertices of S2 are not S2 -saturated. However, S1 is a minimal connected global defensive (−1)-alliance. Furthermore, from S2 (which is a global defensive 0-alliance), we obtain a minimal connected global defensive (−1)-alliance S1 by saturating the vertex b (i.e. by making the vertex c out of the alliance S2 ). Thus, S1 contains one S1 -saturated vertex ‘‘b’’ and one not S1 -saturated vertex ‘‘a’’. On the other hand, S1 is more sensitive than S2 in the sense that ‘‘adding a new white vertex adjacent to the S1 -saturated vertex b’’ or ‘‘removing in ES1 an adjacent edge to b’’ in the graph will necessarily change the structure of the alliance. In other words, adding a new vertex or removing an edge can disrupt the alliance, which makes its stability fragile and at risk. Thus, because of their sensitive situation, it is interesting to consider alliances of this type and to study their properties. It is clear that every defensive (−1)-alliance S can have S-saturated vertices and not S-saturated vertices. From this, we give the following definition.

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Definition 6. Let S be a defensive (−1)-alliance of a given graph G = (V , E ). We define λ(S ) as the set of S-saturated ¯ S ) as the set of not S-saturated vertices of S, i.e. vertices of S, i.e. λ(S ) = {v ∈ S : dS (v) + 1 = dV −S (v)}, and λ( ¯ S ) = {v ∈ S : dS (v) + 1 > dV −S (v)}. λ( Remark 1. We observe what follows:

• • •

¯ S )| = |S |. Clearly, we have |λ(S )| + |λ( |λ(S )| = |S | if and only if the defensive (−1)-alliance S is saturated. ¯ S )| = |S | if and only if the defensive (−1)-alliance S is strong. |λ(

¯ S )) as a set of S-saturated (resp. not S-saturated) vertices of ∂ S Remark 2. In the same way, we can define λ(∂ S ) (resp. λ(∂ for an offensive 1-alliance S. In the following lemma, we prove that among all the boundary k-alliances, the only ones that are saturated are those that correspond to k = −1 (or k = 1). Lemma 1. A boundary defensive (resp. offensive) k-alliance is saturated if and only if k = −1 (resp. k = 1). Proof. Let S ⊆ V be a boundary defensive k-alliance in G = (V , E ). Then dS (v) = dV −S (v) + k,

∀ v ∈ S.

(2)

∀ v ∈ S.

(3)

If S is saturated, then dS (v) + 1 = dV −S (v),

Hence, by (2) and (3), we obtain that k = −1. Conversely, if k = −1, from (2) we obtain dS (v) = dV −S (v) − 1, ∀ v ∈ S and it follows that S is saturated. In a similar way we can prove the rest of this lemma.  In the following result, we give sufficient conditions for a dominating set to be a minimal global boundary defensive

(−1)-alliance by using the concept of S-saturated vertices.

Lemma 2. Let G = (V , E ) be a graph. If there exists a dominating set S ⊂ V such that for all v ∈ S , v is S-saturated, then S is a minimal global boundary defensive (−1)-alliance. Proof. Since for all v ∈ S ⊂ V , v is an S-saturated vertex, then

∀ v ∈ S,

dS (v) = dV −S (v) − 1.

As S is a dominating set, it follows that S is a global boundary defensive (−1)-alliance. Now, we show that S is minimal. We proceed by contradiction. Suppose that S, |S | = ρ , is not minimal. Then there exists S˜ ⊂ S , |S˜ | < ρ , such that S˜ is a global boundary defensive (−1)-alliance. ˜ then we have N (x) ∩ S˜ = ∅ or N (x) ∩ S˜ ̸= ∅: Let x ∈ S − S,

˜ A contradiction with S˜ is a global boundary defensive (−1)-alliance. 1. If N (x) ∩ S˜ = ∅, then x will be not dominated by S. ˜ ˜ 2. If N (x) ∩ S ̸= ∅, then there exists y ∈ N (x) ∩ S such that |N [y] ∩ S˜ | < |N [y] ∩ S | = |N [y] ∩ (V − S )| < |N [y] ∩ (V − S˜ )|. Thus,

|N [y] ∩ S˜ | < |N [y] ∩ (V − S˜ )| and then dS˜ (y) + 1 < dV −S˜ (y), that is a contradiction with S˜ is a global boundary defensive (−1)-alliance. Thus, we conclude that S is a minimal global boundary defensive (−1)-alliance.  From Lemma 2, we obtain the following corollary. Corollary 1. Let G = (V , E ) be a graph and suppose that S ⊂ V is a global defensive (−1)-alliance. If S is not minimal, then ¯ S )| ̸= 0). there exists at least one vertex which is not S-saturated (i.e. |λ(

¯ S ), ES and ω(S ) in the case where S is a defensive (−1)In the following lemma, we obtain a relation between λ(S ), λ( alliance. ¯ S )|. Lemma 3. If S is a defensive (−1)-alliance of a given graph G = (V , E ), then |λ(S )| ≥ |ω(S )| − 2|ES | − |λ( Proof. Since S is a defensive (−1)-alliance, then

∀ v ∈ S,

dS (v) ≥ dV −S (v) − 1.

(4)

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By summing the relation (4), we obtain



dS (v) ≥

v∈S

 (dV −S (v) − 1), v∈S

and it follows that 2|ES | ≥ |ω(S )| − |S |.

¯ S )|, the conclusion follows. Thus, by replacing |S | by |λ(S )| + |λ(



As particular case of Lemma 3, if we have a boundary defensive (−1)-alliance, we obtain the following lemma. Lemma 4. If S is a boundary defensive (−1)-alliance of a given graph G = (V , E ), then |λ(S )| = |S | = |ω(S )| − 2|ES |. Remark 3. Note that Lemma 4 is also a particular case of Theorem 4 (i), for k = −1, given in [26]. In the same way as in Lemma 4, we can prove the following result. Lemma 5. If S is a global boundary offensive 1-alliance of a given graph G = (V , E ), then |ω(S )| = 2|EV −S | + |V − S |. From Lemmas 2 and 4, we obtain Lemma 6. Let G = (V , E ) be a graph. If there exists a dominating set S ⊂ V such that for all v ∈ S , v is S-saturated, then S is a minimal global boundary defensive (−1)-alliance and |S | = |ω(S )| − 2|ES |. 4. Global boundary powerful (−1)-alliances In this section, we study mathematical properties of global boundary powerful (−1)-alliances and we obtain theoretical results by giving in particular some bounds and exact values on the cardinality of such alliances. Let us start with the following definition. Definition 7. Let S be a powerful (−1)-alliance of a given graph G = (V , E ). We say that S is a boundary powerful (−1)alliance (resp. global boundary powerful (−1)-alliance), if every vertex of N [S ] = S ∪ ∂ S (resp. V ) is S-saturated. For the second case, we obtain λ(S ) ∪ λ(∂ S ) = V . Remark 4. (i) From Lemma 1, we get: a boundary powerful k-alliance is saturated if and only if k = −1. (ii) A boundary powerful (−1)-alliance S ⊆ V (resp. global) has every vertex of N [S ] (resp. V ) is S-saturated. Thereafter, we use the following notations:

• GBPA: Global boundary powerful (−1)-alliance; • MGBPA: Minimal global boundary powerful (−1)-alliance; • MCGBPA: Minimal connected global boundary powerful (−1)-alliance. Since for every GBPA we have |λ(S )| = |S |, in the results that we present in this section we use |S | instead of |λ(S )|. Now, we start by giving several lemmas that essentially will be used to prove Theorem 1 which is itself useful to prove the main result (Theorem 2) where tight bounds on the cardinality of every MGBPA will be given by considering the connected and not connected cases. From Lemma 2, we get Lemma 7. Let G = (V , E ) be a graph. Every GBPA S ⊂ V is minimal. From Lemmas 4 and 5, we obtain Lemma 8. If S is a GBPA of a given graph G = (V , E ), then |ω(S )| = 2|ES | + |S | = 2|EV −S | + |V − S |. In the following lemma, we give an upper bound for the size of any not connected subgraph of G. Lemma 9. Let G = (V , E ) be a graph and S ⊂ V . If S is not connected, then |ES | ≤ |S |(|S |−1)

|S |2 −3|S |+2

|S |2 −3|S |+2 2

.

|S |(|S |−1)

Proof. If S is not connected, then |ES | ≤ − (|S | − 1) = such that is the number of edges which 2 2 2 can have a clique C of order |S | and |S | − 1 is the minimum number of edges that we should remove from C in order to disconnect it (to disconnect at least one vertex from C ).  Now, we show that for a graph G = (V , E ) if a subset of vertices S is a MGBPA, then necessarily its complementary in V is a MGBPA. Lemma 10. Let G = (V , E ) be a graph and S ⊂ V . S is a MGBPA if and only if V − S is a MGBPA. Proof. Suppose that S is a MGBPA. Since S is a GBPA, then ∀ v ∈ V , |N [v] ∩ S | = |N [v] ∩ (V − S )|. It follows that ∀ v ∈ S ∪(V − S ), |N [v]∩ S | = |N [v]∩(V − S )|. Thus, (V − S ) is a GBPA, and therefore, (V − S ) is a MGBPA by using Lemma 7. The converse implication is a consequence of double complement law. 

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Remark 5. Note that Lemma 10 is an extension of Remark 9(i) [27] for global boundary powerful (−1)-alliances which are minimal. In the following lemma, we prove that the number of vertices of every graph containing MGBPA is even. Lemma 11. Let G = (V , E ) be a graph with |V | = n and |E | = m. If S ⊂ V is a MGBPA, then n = 2(m − 2|ES | − 2|EV −S |). Proof. We have (v) = v∈S dG (v) + v∈(V −S ) dG (v) = 2m. By replacing dG (v) by dS (v) + dV −S (v), we obtain v∈V dG [ d (v)+ d (v)]+ S V − S v∈S v∈(V −S ) [dS (v)+ dV −S (v)] = 2m. From this, it follows that 2|ES |+|ω(S )|+|ω(S )|+ 2|EV −S | = 2m. By using Lemma 8, we get 2|ES |+(2|ES |+|S |)+(2|EV −S |+|V − S |)+ 2|EV −S | = 2m. Thus, 4|ES |+ 4|EV −S |+|S |+|V − S | = 2m and hence n = |S | + |V − S | = 2(m − 2|ES | − 2|EV −S |). 









Remark 6. From Lemma 11, we deduce that the graphs of odd order do not contain MGBPA. Now, we give upper bounds for |S | in the case where S is a MGBPA. Lemma 12. Let G = (V , E ) be a graph with |V | = n and |E | = m. If S ⊂ V is a MGBPA, then: (i) |S | ≤ |V − S |2 , | V −S | (ii) |S | ≤ |V −S |+1 n. Proof. To show the inequalities (i) and (ii), we proceed by contradiction: (i) Suppose that |S | > |V − S |2 . Since S is a global boundary offensive 1-alliance, then ∀ v ∈ (V − S ), dS (v) = dV −S (v)+ 1 ≤ |V − S |. By summing on v , we obtain v∈(V −S ) dS (v) ≤ |V − S |2 , and it follows that ω(S ) = ω(V − S ) ≤ |V − S |2 . By using the hypothesis, we get |S | > ω(S ) = 2|ES | + |S |, and hence 2|ES | < 0 which is a contradiction. | V −S | (ii) Suppose that |S | > |V −S |+1 n. From this, we get |S ||V − S | + |S | > |V − S |n. By replacing n by |S | + |V − S |, we obtain

|S | > |V − S |2 which is a contradiction with (i).



Remark 7. The upper bounds given in Lemma 12 are attained, for example, for K2 (|S | = |V − S | = 1) or for a graph such that V − S is a clique with each vertex of V − S having a degree in S equal to |V − S | and every vertex of S is a leaf. From Lemma 12, we obtain the following other bounds. Lemma 13. Let G = (V , E ) be a graph with |V | = n and |E | = m. If S ⊂ V is a MGBPA, then: √

(i) |S | ≤ 2n+1−2 4n+1 , √ (ii) n ≥ |S | + |S |. Proof. (i) From Lemma 12 (ii), we have |S | ≤ |V −S |+1 n and then |S |(|V − S |+ 1)− n|V − S | ≤ 0. By replacing |V − S | by n −|S |, we get |S |(n − |S √ | + 1) − n(n − |S |) ≤ 0√and it follows that |S |2 − (2n + 1)|S | + n2 ≥√ 0. Thus, |S | ∈ ] − ∞, x1 ] ∪ [x2 , +∞[ |V −S |

with x1 =

2n+1− 4n+1 2

2n+1+ 4n+1 . Since 2 2n+1− 4n+1 and then 4n 2

and x2 = √

√

x2 > n, then |S | ≤ x1 =

2n+1− 4n+1 . 2

(ii) From (i), we have |S | ≤ + 1 ≤ 2n + 1 − 2|S |. From this, we get n2 − 2|S |n + |S |(|S | − 1) ≥ 0. √ √ √ Thus, n ∈ ] − ∞, n1 ] ∪ [n2 , +∞[ with n1 = |S | − |S | and n2 = |S | + |S |. We conclude that n ≥ n2 = |S | + |S |.  In the following theorem, we give different bounds by considering connected and non-connected MGBPA. These bounds will be useful to prove the main result (Theorem 2). Theorem 1. Let G = (V , E ) be a graph with |V | = n and |E | = m. If S ⊂ V is a MGBPA, then:

• If S is connected, we have: √ (i) |S | + |S | ≤ n ≤ |S |(|S | + 1), (ii) 4|S | − 3 ≤ m ≤ |K2|S | |. • If S is not connected, √ the inequalities become: (iii) Max{|S | + |S |, 2Min{|S |, |V − S |}} ≤ n ≤ |S |(|S | − 1) + 2, (iv) |S | ≤ m ≤ |K2(|S |−1) | + 1. Proof. To show the inequalities (i)–(iv), we proceed by contradiction:

√

(i) • n ≥ |S | + |S | is given by Lemma 13(ii). • Suppose that n > |S |(|S |+1). From this, we obtain n−|S | > |S |2 and therefore |V −S | > |S |2 . Since the vertices of V −S are dominated (S is global), then |ω(S )| ≥ |V − S | > |S |2 . By Lemma 8, we obtain 2|ES | + |S | > |S |2 and it follows that |ES | > |S |(|S2|−1) which is a contradiction, because |S |(|S2|−1) is the maximum of edges that ES can have when S is a clique. (ii) • Suppose that 4|S | − 3 > m. By replacing m by |ES | + |EV −S | + |ω(S )| and using Lemma 8, we obtain 4|S | − 3 > |ES | + |EV −S | + (2|ES | + |S |). Thus, it follows that 3|S | − 3 > 3|ES | + |EV −S |. Since S is connected (|ES | ≥ |S | − 1), then 3|S | − 3 > 3(|S | − 1) + |EV −S |. Hence, 0 > |EV −S | and this is a contradiction.

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• Suppose that m > |K2|S | |. This gives m > 2|S |(22|S |−1) and then |ES | + |EV −S | + |ω(S )| > 2|S |2 − |S |. By using Lemma 8, |S |(|S |−1) we obtain |ES | + |EV −S | + (2|ES | + |S |) > 2|S |2 − |S | and then 3|ES | + |EV −S | > 2|S |2 − 2|S |. Since |ES | ≤ , we get 2 |S |(|S |−1) |S |(|S |−1) |V −S |(|V −S |−1) 2 3( ) + |EV −S | > 2|S | − 2|S |. From this, we deduce that |EV −S | > ≥ |ES |. Since |EV −S | ≤ , 2 2 2 we obtain

|V − S |(|V − S | − 1) 2

≥ |EV −S | >

|S |(|S | − 1) 2

≥ |ES |.

(5)

From (5) we have |V − S | > |S |, and by using Lemma 8, from |EV −S | > |ES |, we obtain |S | > |V − S | which is a contradiction. (iii) • Suppose that 2Min{|S |, |V − S |} > n. If |S | ≤ |V − S |, then 2|S | > n. It follows that 2|S | > |S | + |V − S |. Thus, |S | > |V − S | and this is a contradiction. In the same way we obtain a contradiction if |S | ≥ |V − S |.√ On the other hand, from Lemma 13 (ii), we have n ≥ |S | + |S |, and then the conclusion follows. • Suppose that n > |S |(|S | − 1) + 2. By replacing n by |S | + |V − S |, we obtain |S | + |V − S | > |S |2 − |S | + 2 and then |V − S | > |S |2 − 2|S | + 2. Since S is global, then |ω(S )| ≥ |V − S | > |S |2 − 2|S | + 2 which, by using Lemma 8, yields |S |2 −3|S |+2

which is a contradiction to Lemma 9. 2|ES | + |S | > |S |2 − 2|S | + 2. Therefore |ES | > 2 (iv) • Suppose that |S | > m. From this, we obtain |S | > |ES | + |EV −S | + |ω(S )| which, by using Lemma 8, gives |S | > |ES | + |EV −S | + 2|ES | + |S |. Hence, 0 > 3|ES | + |EV −S | and this is a contradiction. • We have seen in (ii) that if S is connected the maximum number of edges is m = |K2|S | |. Starting from the graph K2|S | , to disconnect S it should isolate at least one vertex v of S from the other vertices of S. It follows that, it should remove at least |S | − 1 edges of ES to isolate at least one vertex of S (to disconnect S). Furthermore, to keep the saturation of the vertices adjacent to the removed edges of ES , for every removed edge [v1 , v2 ] ∈ ES , it should remove with it three other edges to form a cycle of length 4 passing through the two vertices v1 and v2 of S and two other vertices v3 and v4 of V − S. In this case, each vertex of {v1 , v2 , v3 , v4 } loses a neighbor of S and a neighbor of V − S. It follows that, every vertex of V is saturated and S is still a minimal (connected) global boundary powerful (−1)-alliance (M(C)GBPA) of the obtained graph. Thus, in total, it should remove at least 4(|S | − 1) edges from K2|S | in order to isolate at least one vertex of S (to disconnect S) and keeping S a M(C)GBPA. Therefore, if S is not connected, m ≤ |K2|S | | − 4(|S | − 1) = 2|S |2 − 5|S | + 4 = |K2(|S |−1) | + 1 and the conclusion follows. 

2|S |(2|S |−1) 2

− 4(|S | − 1) =

Remark 8. Let G = (V , E ) be a graph with |V | = n and |E | = m. If S ⊂ V is a MCGBPA, then (i) • The maximum number of vertices that the graph G can have is reached when the subgraph induced by S is a clique and each vertex of V − S is a leaf. In this case, ∀ v ∈ S , dV −S (v) = |S | then |V − S | = |S |2 and |V | = |S |2 + |S |. • The minimum number of vertices that the graph G can have is reached, for example, when G = K2 . (ii) • The maximum number of edges that the graph G can have is reached when G is a complete graph of order 2|S |, i.e. G = K2|S | . • The minimum number of edges that the graph G can have is reached when the subgraph induced by S is an elementary path and each vertex of V − S is a leaf (i.e. G is a tree). In this case, we have |S | − 2 vertices of S of degree 3 in V − S and 2 vertices of S of degree 2 in V − S. Thus m = |ω(S )| + |ES | + |EV −S | = [3(|S | − 2) + 2 × 2] + [|S | − 1] + 0 = 4|S | − 3. However, if S is not connected, then (iii) • The maximum number of vertices that the graph G can have is reached, for example, when there exists an isolated edge which links a vertex x of S with a vertex y of V − S and the subgraph induced by S˜ = S − {x} is a clique such that each vertex of V˜ = V − {S˜ ∪ {x, y}} is a leaf. In this case, ∀ v ∈ S˜ , dV˜ (v) = |S | − 1 then |V˜ | = (|S | − 1)2 and

|V | = |V˜ | + |S˜ | + 2 = (|S | − 1)2 + (|S | − 1) + 2 = |S |(|S | − 1) + 2. • The minimum number of vertices that the graph G can have is reached, for example, when the graph G is constituted of not adjacent edges and every edge links a vertex of S with a vertex of V − S. (iv) • The maximum number of edges that the graph G can have is reached, for example, when there exists an isolated edge which links a vertex x of S with a vertex y of V − S and the subgraph induced by V −{x, y} is complete of order 2(|S |− 1), i.e. G is composed of two connected components K2(|S |−1) and the edge [x, y]. Thus m = |K2(|S |−1) | + 1. • The minimum number of edges that the graph G can have is |S |. This number is reached when the graph G is constituted of not adjacent edges and every edge links a vertex of S with a vertex of V − S.

From Theorem 1, we obtain the following main result which provides tight bounds for the cardinality of every MGBPA, in terms only of n and m that are the order and the size of graph respectively. Theorem 2. Let G = (V , E ) be a graph with |V | = n and |E | = m. If S ⊂ V is a MGBPA, then:

H. Slimani, H. Kheddouci / Discrete Applied Mathematics 185 (2015) 192–207

199

(i) If S is connected, we have:

 Max

√ −1 +

√

1 + 4n 1 +

,

2



1 + 8m

 ≤ |S | ≤ Min

4

√ 2n + 1 −

4n + 1 m + 3

,

2

 .

4

(6)

(ii) If S is not connected, the relation becomes:

 Max

√ 1+

√

4n − 7 5 +

8m − 7

2

4

,



 ≤ |S | ≤ Min

√ 2n + 1 −

4n + 1

2

 ,m .

(7)

Proof. From Theorem 1, we obtain: (i) If S is connected, we have: √ • |S | + |S | ≤ n ≤ |S |(|S | + 1), √ ◃ From |S |+ |S | ≤ n, we obtain |S |2 −(2n + 1)|S |+ n2 ≥ 0. Thus |S | ∈ ]−∞, x1 ]∪[x2 , +∞[ with x1 = √

√

√ 2n+1− 4n+1 2

and x2 = 2n+1+2 4n+1 . Since x2 > n, then |S | < x1 = 2n+1−2 4n+1 . Thus we find the upper bound of |S | given in Lemma 13 (i). √ −1− 1+4n and ◃ From |S |(|S | + 1) ≥ n, we obtain |S |2 + |S | − n ≥ 0. Thus, |S | ∈ ] − ∞, y1 ] ∪ [y2 , +∞[ with y1 = 2 √

√

y2 = −1+ 2 1+4n . Since y1 < 0 and |S | ≥ 0, then |S | ≥ y2 = −1+ 2 1+4n . Thus, we conclude that:

√ −1 +

√ 1 + 4n

≤ |S | ≤

2n + 1 −

4n + 1

2 2 • 4|S | − 3 ≤ m ≤ |K2|S | |, ◃ From 4|S | − 3 ≤ m, we deduce that |S | ≤ ◃ From m ≤ |K2|S | | = √

2|S |(2|S |−1) 2

.

(8)

m+3 . 4

= |S |(2|S | − 1), we obtain 2|S |2 − |S | − m ≥ 0. Thus, |S | ∈ ] − ∞, z1 ] ∪ [z2 , +∞[ √

with z1 = 1− 41+8m and z2 = 1+ Thus, we conclude that:

1+8m . 4

Since z1 < 0 and |S | ≥ 0, then |S | ≥ z2 = 1+

√

1+8m . 4

√ 1 + 8m

1+

≤ |S | ≤

m+3

.

(9)

4 4 Therefore, from (8) and (9), we obtain the relation (6). (ii) If S is not connected, we have: √ • Max{|S | + |S |, 2Min{|S |, |V − S |}} ≤ n ≤ √|S |(|S | − 1) + 2, 2n+1− 4n+1 ◃ From Lemma 13 (i), we have |S | ≤ . 2 ◃ From |S |(|S | − 1) + 2 ≥ n, we obtain |S |2 − |S | + 2 − n ≥ 0. Thus, |S | ∈ ] − ∞, a1 ] ∪ [a2 , +∞[ with a1 = √

and a2 = 1+

4n−7 2 √

√ 1− 4n−7 2

. Since a1 < 0 and |S | ≥ 2 (S contains at least two vertices because it is not connected), then

1+ 4n−7 . 2

|S | ≥ a2 = Thus, we conclude that: √ 1+

√ 4n − 7

2

≤ |S | ≤

2n + 1 −

• |S | ≤ m ≤ |K2(|S |−1) | + 1, ◃ We have |S | ≤ m. ◃ From m ≤ |K2(|S |−1) | + 1 =

2

.

(10)

2(|S |−1)[2(|S |−1)−1] 1 2 √ 5− 8m−7 with b1 4

|S | ∈ ] − ∞, b1 ] ∪ [b2 , +∞[ √

4n + 1

+

=

= 2|S |2 − 5|S | + 4, we obtain 2|S |2 − 5|S | + 4 − m ≥ 0. Thus, √

and b2 = 5+

8m−7 . 4

Since b1 ≤ 1, ∀ m ≥ 1 and |S | ≥ 2, then

5+ 8m−7 . 4

|S | ≥ b2 = Thus, we conclude that: √ 5+

8m − 7 ≤ |S | ≤ m. 4 Therefore, from (10) and (11), we obtain the relation (7).

(11) 

Remark 9. Note that the bounds given in Theorem 2 are tight and hold for the graphs given in Fig. 3 which contain M(C)GBPA (for each graph, the black vertices represent the alliance). The bounds attained are given in bold as illustrated in Table 1. Note that for G3 = K2 , all the bounds given in Theorem 2(i) are attained. 5. Procedures for generating graphs containing M(C)GBPA In order to generate graphs containing M(C)GBPA, we need to define some procedures of construction.

200

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Fig. 3. Graphs containing M(C)GBPA.

Fig. 4. Procedures P1 and P2 . Table 1 Bounds of Theorem 2 for G1 –G8 . Graph

Bounds of Theorem 2 (i), inequalities (6) √ −1+ 1+4n 2

√

√

1+ 1+8m 4

|S |

2n+1− 4n+1 2

m+3 4

3 2.71 1 2.38

3 3 1 3

4 9 1 7.29

4.5 3.75 1 3

G1 G2 G3 G4

2 3 1 2.70

Graph

Bounds of Theorem 2 (ii), inequalities (7) √

G5 G6 G7 G8

√

√

1+ 4n−7 2

5+ 8m−7 4

|S |

2n+1− 4n+1 2

m

2.56 3 2.56 2.56

3 2.85 2.28 2.68

3 3 3 4

4 5.62 4 4

7 6 3 5

Definition 8. We define the procedures P1 and P2 as procedures to apply to a K4 to obtain graphs containing M(C)GBPA with the same number of vertices of S, given in Fig. 4. Lemma 14. Let G = (V , E ) be a graph containing a MGBPA S ⊂ V . For every K4 of G, composed of two vertices of S and two other vertices of V − S, if we apply to it the procedure P1 or P2 we obtain a new graph where S is still a MGBPA. Proof. For a given K4 of G, composed of two vertices of S (a and b) and two other vertices of V − S (c and d):

• if we apply to it the procedure P1 , we obtain a new graph with two new vertices (e, f ∈ V − S) where each vertex of {c , d} loses a neighbor of S and a neighbor of V − S, the vertex a (resp. b) loses a neighbor of V − S which is replaced by the vertex e (resp. f ) and the neighborhood of each vertex of V − X with X = {a, b, c , d} remains unchangeable. Thus every vertex of V ∪ {e, f } is S-saturated and S is still a MGBPA of the obtained graph. • if we apply to it the procedure P2 , each vertex of X loses a neighbor of S and a neighbor of V − S and the neighborhood of each vertex of V − X remains unchangeable. Thus every vertex of V is S-saturated and S is still a MGBPA of the obtained graph.



Remark 10. Let G = (V , E ) be a graph containing a MCGBPA S ⊂ V . Let K4 be a subgraph of G composed of two vertices of S and two other vertices of V − S. We denote the edge which links the vertices of S in the K4 by u. Then we have:

• By applying the procedure P1 to the K4 , we get a graph with S that is still connected. • By applying the procedure P2 to the K4 : If u belongs to a cycle of ES , we get a graph with S that is still connected. If u does not belong to a cycle of ES , we get a graph with S that is not connected. In addition to the procedures P1 and P2 , which keep the number of vertices of S unchangeable after their applications to a given K4 , we can define other procedures that increase the number of vertices of S.

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201

Fig. 5. K4 .

Fig. 6. Procedures to obtain M(C)GBPA from a given K4 .

As in the procedure P1 , from a given K4 we remove some edge(s) then we add other vertices and edges in order to obtain a new graph containing a M(C)GBPA. However, in some cases it will be necessary to add black vertices which makes increasing the number of vertices of S in the new graph. Thus, for any vertex x ∈ V (black or white), if we remove two adjacent edges of type (x, y) and (x, z ) with y a black vertex and z a white vertex, then x still remains an S-saturated vertex without adding no edge and no vertex. However, if we remove one edge of type (x, t ), to make the vertex x S-saturated we should add a new vertex v which is a leaf (of the same color of t) adjacent to x. We add as many times as necessary vertices of type v corresponding to the number of edges of type (x, t ) removed. Thus, from a given K4 presented in Fig. 5, we have the following different situations: 1. If we remove the edge 1, we should remove with it (2 or 6) and (3 or 5) in order not to add a black vertex adjacent to another black one (a or b). Otherwise, the alliance becomes a not boundary defensive (−1)-alliance. 2. If we remove the edge 4, we should remove with it (3 or 6) and (2 or 5) in order not to add a white vertex adjacent to another white one (c or d). Otherwise, the alliance becomes a not boundary offensive 1-alliance and not global. 3. If we want remove one edge, we can remove the edge 2 or 3 or 5 or 6 and the resulting graph is the same for each one. 4. If we want remove two edges, we have two different possibilities. We can remove two adjacent edges or two not adjacent edges among {2, 3, 5, 6}, i.e. ‘‘{2, 6} or {3, 5}’’ and ‘‘{2 or 6} with {3 or 5}’’. 5. If we want remove three edges, we have three different possibilities. We can remove ‘‘1 and {2 or 6} and {3 or 5}’’ or ‘‘4 and {3 or 6} and {2 or 5}’’ or ‘‘three among {2, 3, 5, 6}’’. 6. If we want remove four edges, we have two different possibilities. We can remove ‘‘{1, 4} with {{2, 3} or {5, 6}}’’ or ‘‘{2, 3, 5, 6}’’. Thus, we define the following different eight procedures by including the procedures P1 and P2 already given in Definition 8: P1 : is the procedure which consists to remove from a K4 the edges 2, 3 and 4 and add two white vertices and two edges to obtain the graph (4) given in Fig. 6. P2 : is the procedure which consists to remove from a K4 the edges 1, 2, 3 and 4 to obtain the graph (7) given in Fig. 6. P3 : is the procedure which consists to remove from a K4 the edge 2 (that links a black vertex with a white one) and add two vertices (one black and one white) and two edges to obtain the graph (1) given in Fig. 6. P4 : is the procedure which consists to remove from a K4 the edges 2 and 3 and add four vertices (two black and two white) and four edges to obtain the graph (2) given in Fig. 6. P5 : is the procedure which consists to remove from a K4 the edges 2 and 6 and add four vertices (two black and two white) and four edges to obtain the graph (3) given in Fig. 6. P6 : is the procedure which consists to remove from a K4 the edges 1, 2, and 3 and add two black vertices and two edges to obtain the graph (5) given in Fig. 6. P7 : is the procedure which consists to remove from a K4 the edges 2, 3 and 5 and add six vertices (three black and three white) and six edges to obtain the graph (6) given in Fig. 6. P8 : is the procedure which consists to remove from a K4 the edges 2, 3, 5 and 6 and add eight vertices (four black and four white) and eight edges to obtain the graph (8) given in Fig. 6.

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6. Algorithms generating graphs containing M(C)GBPA Using the procedures P1 –P8 , we establish the following algorithms which generate graphs containing M(C)GBPA from a given complete graph K2n = (V , E ), with n > 1. 6.1. Algorithm generating graphs containing MCGBPA following an order of edges of ES The following algorithm generates different graphs containing MCGBPA by using a given order of edges of ES . For i ≥ 0 and j ≥ 1, we denote:

• K4(j) : a K4 composed of the vertices of the j-th edge in the given order of edges of ES and 2 other vertices of (V − S ); (i,j) • K2n : the graph containing MCGBPA obtained from K2n at the order (i, j), for j > i and j > 1, where i designates the (j−1) (0,1) number of edges removed from ES and j corresponds to the order of graph obtained by using K4 . We set K2n := K2n . Algorithm 1 Algorithm generating graphs containing MCGBPA following an order of edges of ES Input: n ∈ N with n > 1. Output: Frn : set of graphs containing MCGBPA corresponding to an order of edges of ES . 1: Step 0: Choose S ⊂ V such that |S | = n. S is a MCGBPA in K2n ; 2: Step 1: Set i := 0; (0,1) 3: Step 2: Set |ES | := |Kn | − i = ki . Number the edges of ES in K2n := K2n from 1 to ki such that the partial subgraph induced by S and the last (n − 1) edges of ES is connected. We denote this order by σr (ES ). 4: Step 3: Set j := i + 1; 5: Step 4: Determination of a graph containing MCGBPA which has ki edges: (i,j) (j) • Form in the graph K2n a complete graph K4 composed of the 2 vertices of the j-th edge in σr (ES ) (that we denote by aj and bj ) and 2 other vertices of V − S (that we denote by xj and yj ); (i,j) • Remove in K2n , the edges (bj , yj ), (yj , xj ) and (xj , aj ) and add 2 vertices cj , dj ∈ (V − S ) and 2 edges (bj , cj ) and (aj , cj ) (j)

6: 7: 8: 9: 10: 11:

(i,j+1)

(i.e. the procedure P1 is applied to the K4 ); Thus, S is a MCGBPA in the obtained graph K2n ; Step 5: if j < |Kn | then Set j := j + 1 and go to Step 4; else if |ES | > n − 1 then (i,j) Set j := i + 1. Remove in K2n the edges j = (aj , bj ), (bj , yj ), (yj , xj ) and (xj , aj ) (i.e. remove the cycle (aj , bj , yj , xj , aj ) (j)

12: 13:

(i+1,j+1)

by applying the procedure P2 to the K4 ). Thus, S is a MCGBPA which has ki − 1 edges in the obtained graph K2n . Set i := i + 1, |ES | := |Kn | − i = ki and go to Step 3; else It is the end of the algorithm, the set of graphs containing MCGBPA corresponding to the order σr (ES ) is Frn = (i,j) {(K2n , j = i + 1, ..., |Kn | + 1), i = 0, ..., |Kn | − (n − 1)}.

14: 15:

end if end if

Example 2. In this example, we give the different graphs containing MCGBPA obtained from K6 by applying Algorithm 1 (see Fig. 7). Proposition 1. The number of different graphs containing MCGBPA that can be obtained from the complete graph K2n with n > 1, by using Algorithm 1 for a given order σr (ES ), is N =

n4 −2n3 +3n2 −2n+8 . 8

Proof. For n > 1, the graphs containing MCGBPA that can be obtained from the complete graph K2n = (V , E ) are classified according to the number |ES |. Since the alliances are connected, the number |ES | varies from |Kn | to |S | − 1 with |Kn | (resp. |S | − 1) is the maximum (resp. minimum) number of edges that S can contain. Thus, following Algorithm 1, we have:

|Kn | + 1 |Kn | |Kn | − 1 |Kn | − 2 .. .

|Kn | − (p − 1)

graphs containing MCGBPA with graphs containing MCGBPA with graphs containing MCGBPA with graphs containing MCGBPA with

.. .

graphs containing MCGBPA with

|ES | = |Kn |, |ES | = |Kn | − 1, |ES | = |Kn | − 2, |ES | = |Kn | − 3, .. .

|ES | = |Kn | − p = |S | − 1.

H. Slimani, H. Kheddouci / Discrete Applied Mathematics 185 (2015) 192–207

203

Fig. 7. Graphs containing MCGBPA obtained from K6 by applying Algorithm 1. n(n−1)

2

From the last equality, we obtain p = |Kn | − |S | + 1 = 2 − n + 1 = n −23n+2 . Thus, the total number of different graphs containing MCGBPA that can be obtained is N = |Kn | + 1 + |Kn | + |Kn | − 1 + |Kn | − p p(p+1) +(p + 1) = 2 +· · ·+|Kn |−(p − 1) = k=0 (|Kn |− k + 1) = (p + 1)|Kn |−(0 + 1 + 2 +· · ·+ p)+(p + 1) = (p + 1)|Kn |− 2

(p + 1)[|Kn | −

p 2

+ 1]. By replacing p by

n2 −3n+2 2

and |Kn | by

n(n−1) , 2

we obtain that N =

n4 −2n3 +3n2 −2n+8 . 8



δ

Remark 11. • F = r =1 Frn is the set of graphs containing MCGBPA corresponding to all the orders of edges of ES , where δ = [|Kn | − (n − 1)]!(n − 1)!Nt is the total number of these orders with Nt the number of different spanning trees of Kn (that span S). (i,j) (j) • At Step 4 of Algorithm 1, we can consider in the graph K2n all the complete graphs K4 composed of the 2 vertices of the edge j and 2 other vertices of V − S. In this case, we generate more graphs containing MCGBPA where some of them will be generated several times (with repetition). • On the other hand, at the same Step 4 of Algorithm 1, we can apply to each K4(j) the procedures P1 and P2 instead of P1 alone. Thus, we generate a double number of graphs containing MCGBPA where also some of them will be generated several times (with repetition). The following proposition is obvious. Proposition 2. Let (V , E ) = K2n be a complete graph with n ∈ N\{0}. For all S ⊂ V with |S | = n, S is a MCGBPA. There exist a n number C2n of MCGBPA (of the same type) in K2n . 6.2. Algorithm generating graphs containing M(C)GBPA from K2n , with n > 1 The following algorithm generates graphs containing minimal global boundary powerful (−1)-alliances which are not necessarily connected (M(C)GBPA) from a given complete graph K2n = (V , E ) with n > 1. Unlike Algorithm 1, the following does not use an order of edges of ES . For i, j ≥ 1, we denote:

• • • •

[i,j]

K2n : the i-th graph containing M(C)GBPA obtained from K2n at the order j;

[i,j]

k(i, j): the number of K4 composed of 2 vertices of S and 2 other vertices of V − S that exists in the graph K2n ; [i,j]

8k(i, j): the number of graphs containing M(C)GBPA obtained from the graph K2n by using the procedures P1 –P8 ; k[j]: the number of graphs containing M(C)GBPA obtained at the order j.

Starting from a given complete graph K2n , the principle of the algorithm consists in applying successively the procedures P1 –P8 on every K4 composed of 2 vertices of S and 2 other vertices of V − S of a graph to obtain other graphs containing M(C)GBPA. We repeat the operation on all the obtained graphs until obtaining graphs without K4 . However, at the beginning, [1,1] from the graph K2n = K2n , for each chosen K4 , by applying to it the procedures P1 –P8 , we obtain eight graphs of the same [1,2]

[8,2]

[1,1]

type that K2n to K2n . Thus, it is not necessary to determine and use all the K4 of K2n = K2n one of them (see Algorithm 2).

but it is sufficient to choose

Example 3. In this example, we give the different graphs containing M(C)GBPA obtained from K6 by applying the first iteration of Algorithm 2. Thus from K6 , we obtain the eight graphs containing M(C)GBPA given in Fig. 8. These graphs are obtained by using the procedures P1 –P8 applied to a given subgraph K4 of K6 . Under each graph is indicated its name and the procedure applied to obtain it. We can continue the application of Algorithm 2 by applying the procedures P1 –P8 on all the subgraphs K4 of any obtained graph, until obtaining graphs without subgraphs K4 .

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H. Slimani, H. Kheddouci / Discrete Applied Mathematics 185 (2015) 192–207

Fig. 8. Graphs containing M(C)GBPA obtained from K6 by applying the first iteration of Algorithm 2.

Algorithm 2 Algorithm generating graphs containing M(C)GBPA from K2n Input: n ∈ N with n > 1. Output: F n : set of graphs containing M(C)GBPA obtained from K2n . 1: 2: 3:

[1,2]

4: 5: 6: 7: 8: 9:

10: 11: 12: 13: 14: 15: 16:

[1,1]

Step 0: Choose S ⊂ V such that |S | = n. S is a MCGBPA in K2n = K2n . Set k[1] := 1; Step 1: Set i := 1 and j := 1; [i,j] Step 2: Determine in K2n a K4 composed of 2 vertices of S and 2 other vertices of V − S. By applying the procedures [8,2]

P1 –P8 for the chosen K4 , we obtain the eight graphs K2n to K2n which contain M(C)GBPA. Set k[2] := 8; Step 3: if n > 2 then for j := 2 to |Kn | do Set i := 1; k[j + 1] := 0; while i ≤ k[j] do [i,j] Determine in K2n all the K4 composed of 2 vertices of S and 2 other vertices of V − S. Suppose that there exists a number k(i, j) of K4 . By applying the procedures P1 –P8 for each K4 , we obtain in total 8k(i, j) graphs containing M(C)GBPA. Set k[j + 1] := k[j + 1] + 8k(i, j); i := i + 1; end while end for else go to Step 4. end if [i,j] Step 4: It is the end of the algorithm, F n = {(K2n , i = 1, ..., k[j]), j = 1, ..., |Kn |}.

Remark 12. Note that in Algorithm 2, if we use just the procedures P1 and P2 instead of P1 –P8 , we obtain only graphs containing M(C)GBPA with |S | = n. Example 4. In this example, we give the graphs containing M(C)GBPA obtained from K6 by applying Algorithm 2 (with just the procedures P1 and P2 ). Note that some graphs are generated several times (see Fig. 9). Proposition 3. The number of different graphs containing M(C)GBPA that can be obtained from the complete graph K2n with

¯ = n > 1, by using Algorithm 2 with just the procedures P1 and P2 , is M ≥ M

n4 −2n3 +7n2 −6n+8 . 8

¯ which is the number of graphs containing M(C)GBPA that can be obtained by applying Proof. We determine the number M a modified variant of Algorithm 1. This variant is obtained by effecting slight modifications in Steps 2 and 5 of Algorithm 1 as follows: in Step 2, we number the edges of ES in K2n from j = 1 to j = ki without requiring that the partial subgraph induced by S and the last (n − 1) edges of ES to be connected; in Step 5, instead of testing if |ES | > n − 1, we test if |ES | > 1. Thus, as in the proof of Proposition 1, for n > 1, the graphs containing M(C)GBPA that can be obtained from the complete graph K2n = (V , E ) are classified according to the number |ES |. Since the alliances can be connected or not connected, the number |ES | varies from |Kn | to 0 with |Kn | (resp. 0) is the maximum (resp. minimum) number of edges that S can contain. Therefore, we obtain: |Kn | + 1 |Kn | |Kn | − 1 |Kn | − 2 .. .

.. .

1

graph containing M(C)GBPA with

graphs containing M(C)GBPA with graphs containing M(C)GBPA with graphs containing M(C)GBPA with graphs containing M(C)GBPA with

|ES | = |Kn |, |ES | = |Kn | − 1, |ES | = |Kn | − 2, |ES | = |Kn | − 3, .. . |ES | = 0.

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205

Fig. 9. Graphs containing M(C)GBPA obtained from K6 by applying Algorithm 2 (with just the procedures P1 and P2 ).

¯ = |Kn | + 1 + |Kn | + |Kn | − 1 + |Kn | − 2 + Thus, the total number of graphs containing M(C)GBPA that can be obtained is M |Kn | 2 · · · + 1 = k=0 (|Kn | − k + 1) = (|Kn | + 1) − (0 + 1 + 2 + · · · + |Kn |) = (|Kn | + 1)2 − ( |Kn2|+1 |Kn |). By replacing |Kn | by n(n−1) , 2

4

3

2

¯ = n −2n +7n −6n+8 . we obtain that M 8 Since in Algorithm 2, apart K2n , for each obtained graph we use all the K4 and we apply them the procedures P1 and P2 , contrary to the modified variant of Algorithm 1 where for each obtained graph we use one K4 by following a given order of ¯ and the conclusion follows.  edges of ES . Therefore M ≥ M ¯ = 10. By applying Algorithm 2, some of these graphs are generated several times Remark 13. Note that for n = 3, M = M as given in Example 4. However, the different ten graphs containing M(C)GBPA are those given below, which are classified according to the number |ES | as in the proof of Proposition 3 (see Fig. 10).

206

H. Slimani, H. Kheddouci / Discrete Applied Mathematics 185 (2015) 192–207

Fig. 10. Different graphs containing M(C)GBPA obtained from K6 .

Example 5. In the following table, for some values of n, we give the corresponding numbers N (given in Proposition 1) and ¯ of graphs containing M(C)GBPA that can be obtained. M n

2

3

4

5

10

N ¯ M

2 3

7 10

22 28

56 66

1036 18 146 1081 18 336

20

50 750 926 752 151

Remark 14. Let G = (V , E ) be a graph and S ⊂ V is a M(C)GBPA. In what follows, we enumerate the different cases showing how the structure of the alliance S changes by effecting some disturbances consisting of adding/removing edge(s) or vertex (vertices) to/from the graph G. (1) If we ‘‘add edge(s) connecting vertices of S’’ or ‘‘add a black vertex connecting to S’’ or ‘‘remove a white vertex of degree 1’’, then the (obtained) set S remains a minimal (connected) global boundary offensive 1-alliance (M(C)GBOA) but S becomes a minimal (connected) global defensive (−1)-alliance (M(C)GDA), i.e. S becomes a minimal (connected) global powerful (−1)-alliance (M(C)GPA). (2) If we ‘‘add edge(s) connecting vertices of V − S’’ (resp. ‘‘add a white vertex connecting to V − S’’), then the set S is no longer an offensive 1-alliance but it remains a minimal (connected) global (resp. not global) boundary defensive (−1)-alliance (M(C)GBDA) (resp. M(C)BDA). (3) If we add edge(s) connecting a vertex (vertices) of S with a vertex (vertices) of V − S, then the set S is no longer a defensive (−1)-alliance but S becomes a minimal (connected) global offensive 1-alliance (M(C)GOA). (4) If we ‘‘remove edge(s) connecting vertices of S’’ or ‘‘add a white vertex connecting to a vertex of S’’, then the set S is no longer a defensive (−1)-alliance but it remains a M(C)GBOA. (5) If we ‘‘remove edge(s) connecting vertices of V − S’’ or ‘‘add a black vertex connecting to a vertex of V − S’’, then the (obtained) set S remains a M(C)GBDA but S becomes a M(C)GOA, i.e. S becomes a M(C)GPA. (6) If we remove an edge connecting a vertex of S with a vertex of V − S which is not a leaf (resp. which is a leaf), then S is no longer an offensive 1-alliance but S becomes a M(C)GDA (resp. S becomes a defensive (−1)-alliance which is not global). (7) If we remove a black vertex of degree 1, then the obtained set S˜ ⊂ S is no longer an offensive 1-alliance but it remains a M(C)GBDA (which can be not global if the removed black vertex belongs to an isolated edge). (8) If we remove a white vertex of degree > 1, then the set S becomes a M(C)GDA and a M(C)GOA, i.e. S becomes a M(C)GPA. (9) If we remove a black vertex of degree > 1, then the obtained set S˜ ⊂ S is no longer an offensive 1-alliance and also is no longer a defensive (−1)-alliance, i.e. S˜ is no longer an alliance. From a M(C)GDA (resp. M(C)GOA), we can construct a M(C)GBDA (resp. M(C)GBOA) as follows: (10) If S is a M(C)GDA, we can saturate S to obtain a M(C)GBDA by adding as many as necessary of leaf vertices belonging to V − S and adjacent to the non-saturated vertices of S. (11) If S is a M(C)GOA, we can saturate V − S to obtain a new M(C)GBOA, with probably more black vertices, by adding as many as necessary of edges firstly between non-saturated vertices and secondly if necessary between non-saturated

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Fig. 11. Relations between the different alliances.

vertices and saturated vertices of V − S. If we add edges adjacent to saturated vertices of V − S, these vertices become non-saturated and to re-equilibrate the balance, we add as many as necessary of black leaf vertices adjacent to these created non-saturated vertices of V − S. In Fig. 11, we summarize the relations between the different alliances given in the above remark. 7. Conclusion We have studied mathematical properties of defensive, offensive and powerful alliances when they are in a critical and sensitive situation. The latter represents the limit case when each member has as many bonds in the alliance than outside the alliance (including itself). This means that each member (inside and/or outside the alliance) has as many defenders as attackers. Thus, some disturbance or movement inside or outside the alliance affects necessarily its stability. This can have interpretations in practice especially in the applications related to the security and to the fault tolerance. Acknowledgments The authors are grateful to the anonymous referees for their valuable suggestions and comments which have helped to improve the quality of the paper and its presentation. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

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