Alliance free sets in Cartesian product graphs

Alliance free sets in Cartesian product graphs

Discrete Applied Mathematics 161 (2013) 1618–1625 Contents lists available at SciVerse ScienceDirect Discrete Applied Mathematics journal homepage: ...

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Discrete Applied Mathematics 161 (2013) 1618–1625

Contents lists available at SciVerse ScienceDirect

Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam

Alliance free sets in Cartesian product graphs Ismael G. Yero a , Juan A. Rodríguez-Velázquez b,∗ , Sergio Bermudo c a

Departamento de Matemáticas, Escuela Politécnica Superior de Algeciras, Universidad de Cádiz, Av. Ramón Puyol s/n, 11202 Algeciras, Spain

b

Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Av. Països Catalans 26, 43007 Tarragona, Spain

c

Department of Economy, Quantitative Methods and Economic History, Pablo de Olavide University, Carretera de Utrera Km. 1, 41013-Sevilla, Spain

article

info

Article history: Received 9 December 2011 Received in revised form 7 November 2012 Accepted 4 January 2013 Available online 8 February 2013 Keywords: Defensive alliance Offensive alliance Alliance free set Cartesian product graphs

abstract Let G = (V , E ) be a graph. For a non-empty subset of vertices S ⊆ V , and vertex v ∈ V , let δS (v) = |{u ∈ S: uv ∈ E }| denote the cardinality of the set of neighbors of v in S, and let S = V − S. Consider the following condition:

δS (v) ≥ δS (v) + k,

(1)

which states that a vertex v has at least k more neighbors in S than it has in S. A set S ⊆ V that satisfies Condition (1) for every vertex v ∈ S is called a defensive k-alliance and for every vertex v in the open neighborhood of S is called an offensive k-alliance. A subset of vertices S ⊆ V is a powerful k-alliance if it is both a defensive k-alliance and an offensive (k + 2)alliance. Moreover, a subset X ⊂ V is a defensive (an offensive or a powerful) k-alliance free set if X does not contain any defensive (offensive or powerful, respectively) k-alliance. In this article we study the relationships between defensive (offensive, powerful) k-alliance free sets in Cartesian product graphs and defensive (offensive, powerful) k-alliance free sets in the factor graphs. © 2013 Elsevier B.V. All rights reserved.

1. Introduction The study of relationships between invariants of Cartesian product graphs and invariants of its factor graphs appears frequently in researches about graph theory. In this sense, there are important open problems which are being investigated now. For instance, Vizing’s conjecture [12,13,33], which is one of the most known open problems in graph theory, states that the domination number of the Cartesian product of two graphs is at least equal to the product of the domination numbers of these two graphs. Some variations and partial results about this conjecture have been developed in the last years, like those in [3,5,4,8,31,30]. Apart from the domination number, there are several invariants which have been studied in Cartesian product graphs, for instance, the geodetic number [1,5,10,17], the metric dimension [7], the partition dimension [36], the Menger number [20], the k-domination number [16], the (global) offensive k-alliance number [2,37], the k-alliance partition number [6,35] and the offensive k-alliance partition number [29]. This article concerns the study of alliance free sets in Cartesian product graphs. Since (defensive, offensive and powerful) alliances in graphs were first introduced by Kristiansen et al. [19], several authors have studied their mathematical properties [2,6,15,22,23,27,24–26,28,29,34,35] (the reader is referred to the Ph.D. Thesis [34] for a more complete list of references). Applications of alliances can be found in the Ph.D. Thesis [23] where the author studied the problems of partitioning graphs into alliances and its application to data clustering. On the other hand, defensive alliances represent the



Corresponding author. Tel.: +34 9778511; fax: +34 977558703. E-mail addresses: [email protected] (I.G. Yero), [email protected] (J.A. Rodríguez-Velázquez), [email protected] (S. Bermudo).

0166-218X/$ – see front matter © 2013 Elsevier B.V. All rights reserved. doi:10.1016/j.dam.2013.01.011

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mathematical model of Web communities, by adopting the definition of Web community proposed by Flake et al. in [9], ‘‘a Web Community is a set of web pages having more hyperlinks (in either direction) to members of the set than to nonmembers’’. Other applications of alliances were presented in [14] (where alliances were used in a quantitative analysis of secondary RNA structure), [18] (where alliances were used in the study of predator–prey models on complex networks), [32] (where alliances were used in the study of spatial models of cyclical population interactions) and [21] (where alliances were used as a model of monopoly). In this work we continue the previous studies [23,27,25,26,22] on k-alliance free sets focusing our attention on the particular case of Cartesian product graphs. We study the relationships between defensive (offensive, powerful) k-alliance free sets in Cartesian product graphs and defensive (offensive, powerful) k-alliance free sets in the factor graphs. The plan of the article is as follows: In Section 2 we present the notation and terminology used throughout the paper. Section 3 is devoted to the study of defensive k-alliances. More specifically, we give a sufficient condition for the existence of defensive k-alliance free sets in Cartesian product graphs and we study the relationships between the maximum cardinality of a defensive k-alliance free set in Cartesian product graphs and several invariants of the factor graphs, including the order and the independence number. Analogously, Sections 4 and 5, respectively, are devoted to the study of offensive and powerful k-alliance free sets. In Section 6 we present the conclusions. 2. Notation and terminology In this paper G = (V , E ) denotes a simple graph of order n, minimum degree δ and maximum degree ∆. The independence number of G is denoted by α(G). For a non-empty set S ⊆ V and a vertex v ∈ V , δS (v) denotes the number of neighbors that v has in S and δ(v) denotes the degree of v . The complement of the set S in V is denoted by S. The set of vertices of S which are adjacent to at least one vertex in S is denoted by ∂ S. A non-empty set S ⊆ V is called a defensive (respectively, an offensive) k-alliance in G if for every v ∈ S (respectively, v ∈ ∂ S), δS (v) ≥ δS (v) + k, where k ∈ {−∆, . . . , ∆} (respectively, k ∈ {2 − ∆, . . . , ∆}). Also, a non-empty set of vertices S ⊆ V is called a powerful k-alliance in G if it is both, defensive k-alliance and offensive (k + 2)-alliance, k ∈ {−∆, . . . , ∆ − 2}. Notice that, since V is an offensive k-alliance for every k ∈ {2 − ∆, . . . , ∆}, V is a powerful k-alliance if and only if it is a defensive k-alliance. A set X ⊆ V is (defensive, offensive, powerful) k-alliance free (k-daf, k-oaf, k-paf) if for all (defensive, offensive, powerful) k-alliance S , S \ X ̸= ∅, i.e., X does not contain any (defensive, offensive, powerful) k-alliance as a subset [24]. Associated with the characteristic sets defined above we have the following invariants:

φkd (G): maximum cardinality of a k-daf set in G, k ∈ {−∆, . . . , ∆}. φko (G): maximum cardinality of a k-oaf set in G, k ∈ {2 − ∆, . . . , ∆}. φkp (G): maximum cardinality of a k-paf set in G, k ∈ {−∆, . . . , ∆ − 2}. We now state the following fact on (defensive, offensive and powerful) k-alliance free sets that will be useful throughout this article. Remark 1. If X is a (defensive, offensive, powerful) k-alliance free set and k < k′ , then X is a (defensive, offensive, powerful) k′ -alliance free set. A set Y ⊆ V is a (defensive, offensive, powerful) k-alliance cover set if for all (defensive, offensive, powerful) k-alliance S , S ∩ Y ̸= ∅, i.e., Y contains at least one vertex from each (defensive, offensive, powerful) k-alliance of G. The following duality between k-alliance cover sets and k-alliance free sets allows us to study k-alliance cover sets from the results obtained on k-alliance free sets, so in this article we only consider the study of k-alliance free sets. Remark 2. X is a (defensive, offensive, powerful) k-alliance cover set if and only if X is a (defensive, offensive, powerful) k-alliance free set. We recall that the Cartesian product of two graphs G1 = (V1 , E1 ) and G2 = (V2 , E2 ) is the graph G1 G2 = (V , E ), such that V = {(a, b) : a ∈ V1 , b ∈ V2 } and two vertices (a, b) ∈ V and (c , d) ∈ V are adjacent in G1 G2 if and only if either a = c and bd ∈ E2 or b = d and ac ∈ E1 . For a set A ⊆ V1 × V2 we denote by PVi (A) the projection of A over Vi , i ∈ {1, 2}. 3. Defensive k-alliance free sets in Cartesian product graphs To begin with the study we present the following straightforward result. Remark 3. Let Gi be a graph of order ni , minimum degree δi and maximum degree ∆i , i ∈ {1, 2}. Then, for every k ∈ {1 − δ1 − δ2 , . . . , ∆1 + ∆2 },

φkd (G1 G2 ) ≥ α(G1 )α(G2 ) + min{n1 − α(G1 ), n2 − α(G2 )}.

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Proof. For every graph G of minimum degree δ and maximum degree ∆, any independent set in G is a k-daf set for k ∈ {1 − δ, . . . , ∆}. Hence, φkd (G1 G2 ) ≥ α(G1 G2 ), for every k ∈ {1 − δ1 − δ2 , . . . , ∆1 + ∆2 }, and by the Vizing’s inequality, α(G1 G2 ) ≥ α(G1 )α(G2 ) + min{n1 − α(G1 ), n2 − α(G2 )}, we obtain the result.  Let G1 be the star graph of order t + 1 and let G2 be the path graph of order 3. In this case, φkd (G1 G2 ) = 2t + 1 for k ∈ {−1, 0}. Therefore, the above bound is tight. Even so, Corollary 5 (ii) improves the above bound for the cases where φkd (Gi ) > α(Gi ), for some i ∈ {1, 2}. i

Theorem 4. Let Gi = (Vi , Ei ) be a simple graph of maximum degree ∆i , i ∈ {1, 2}, and let S ⊆ V1 × V2 . Then the following assertions hold. (i) If PVi (S ) is a ki -daf set in Gi , then S is a (ki + ∆j )-daf set in G1 G2 , where j ∈ {1, 2}, j ̸= i. (ii) If for every i ∈ {1, 2}, PVi (S ) is a ki -daf set in Gi , then S is a (k1 + k2 − 1)-daf set in G1 G2 . Proof. Let A ⊆ S and we suppose PV1 (S ) is a k1 -daf set in G1 . Since PV1 (A) ⊆ PV1 (S ), there exists a ∈ PV1 (A) such that δPV (A) (a) < δPV (A) (a) + k1 . If we take b ∈ V2 such that (a, b) ∈ A, then 1

1

δA (a, b) ≤ δPV1 (A) (a) + δPV2 (A) (b) < δPV

1

(A) (a)

+ k1 + δ(b) ≤ δPV

1

(A) (a)

+ k1 + ∆2 .

Thus, A is not a defensive (k1 + ∆2 )-alliance in G1 G2 . Therefore, (i) follows. In order to prove (ii), let x ∈ X = PV1 (A) such that δX (x) < δX (x) + k1 . Let Ax ⊆ A be the set composed of the elements of A whose first component is x. On the other hand, since PV2 (S ) is a k2 -daf set and Y = PV2 (Ax ) ⊆ PV2 (S ), there exists y ∈ Y such that δY (y) < δY (y) + k2 . Notice that (x, y) ∈ A. Let Ay ⊆ A be the set composed of the elements of A whose second component is y. Hence,

δA (x, y) = ≤ < ≤

δAx (x, y) + δAy (x, y) δY (y) + δX (x) δY (y) + δX (x) + k1 + k2 − 1 δAx (x, y) − δ(x) + δAy (x, y) − δ(y) + k1 + k2 − 1

≤ δAx (x, y) + δAy (x, y) + k1 + k2 − 1 = δA (x, y) + k1 + k2 − 1. Thus, A is not a defensive (k1 + k2 − 1)-alliance in G1 G2 and, as a consequence, (ii) follows.



Corollary 5. Let Gl be a graph of order nl , maximum degree ∆l and minimum degree δl , with l ∈ {1, 2}. Then the following assertions hold. (i) For every k ∈ {∆j − ∆i , . . . , ∆i + ∆j }(i, j ∈ {1, 2}, i ̸= j),

φkd (G1 G2 ) ≥ nj φkd−∆j (Gi ). (ii) For every ki ∈ {1 − δi , . . . , ∆i }, i ∈ {1, 2},

φkd +k 1

2 −1

(G1 G2 ) ≥ φkd (G1 )φkd (G2 ) + min{n1 − φkd (G1 ), n2 − φkd (G2 )}. 1

2

1

2

Proof. By Theorem 4(i) we conclude that for every ki -daf set Si in Gi , i ∈ {1, 2}, the sets S1 × V2 and V1 × S2 are (k1 + ∆2 )-daf and (k2 + ∆1 )-daf, respectively, in G1 G2 . Therefore, (i) follows. To construct a (k1 + k2 − 1)-daf set in G1 G2 we will take the product of largest ki -daf sets and we will append to it a largest diagonal. In fact, this idea goes back all the way to Vizing (Theorem 27.17 and Fig. 27.2 in [11]). Let V1 = {u1 , u2 , . . . , un1 }, V2 = {v1 , v2 , . . . , vn2 } and let Si be a ki -daf set of maximum cardinality in Gi , i ∈ {1, 2}. We suppose S1 = {u1 , . . . , ur } and S2 = {v1 , . . . , vs }. By Theorem 4(ii) we deduce that S1 × S2 is a (k1 + k2 − 1)-daf set in G1 G2 . Now let X = {(ur +i , vs+i ), i = 1, . . . , t }, where t = min{n1 − r , n2 − s} and let S = X ∪ (S1 × S2 ). Since, for every x ∈ X , δS (x) = 0 and ki > −δi , i ∈ {1, 2}, we obtain that S is a (k1 + k2 − 1)-daf set in G1 G2 . Thus, φkd +k −1 (G1 G2 ) ≥ |S | = φkd (G1 )φkd (G2 ) + min{n1 − φkd (G1 ), n2 − φkd (G2 )}.  1

2

1

2

1

2

Now we state the following fact that will be useful for an easy understanding of several examples in this paper. Proposition 6. Let G be a graph of order n and maximum degree ∆. Then φkd (G) = n, for each of the following cases: (i) G is a tree of maximum degree ∆ ≥ 2 and k ∈ {2, . . . , ∆}. (ii) G is a planar graph of maximum degree ∆ ≥ 6 and k ∈ {6, . . . , ∆}. (iii) G is a planar triangle-free graph of maximum degree ∆ ≥ 4 and k ∈ {4, . . . , ∆}.

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Fig. 1. This graph is the Cartesian product S3 P4 where S = {(1, 1), (2, 1), (4, 1), (1, 2), (2, 2), (4, 2), (1, 3), (2, 3), (4, 3), (3, 4)} is a maximum defensive 0-alliance free set.

Proof. Suppose S is a defensive k-alliance in G = (V , E ). That is, for every v ∈ S, it follows 2δS (v) ≥ δ(v) + k.

(2)

If some vertex v ∈ S satisfies δ(v) < k, then Eq. (2) leads to δS (v) > δ(v), a contradiction. Hence, for every v ∈ S we have δ(v) ≥ k and, as a consequence, Eq. (2) leads to δS (v) ≥ k. Now, let ms be the size of the subgraph induced by S. Then we have 2ms =



δS (v) ≥ k|S |.

(3)

v∈S

Case (i). Since G is a tree, we obtain 2(|S | − 1) ≥ 2ms ≥ k|S | ≥ 2|S |, a contradiction. For the cases (ii) and (iii) we have |S | ≥ 3, due to that if |S | ≤ 2, then Eq. (2) leads to 2 ≥ δ(v) + k, a contradiction. It is well-known that the size of a planar graph of order n′ ≥ 3 is bounded above by 3(n′ − 2). Moreover, in the case of triangle-free graphs the bound is 2(n′ − 2). Therefore, in case (ii) we have ms ≤ 3(|S | − 2) and, as a consequence, Eq. (3) leads to 6(|S | − 2) ≥ k|S | ≥ 6|S |, a contradiction. Analogously, in case (iii) we have ms ≤ 2(|S | − 2) and, as a consequence, Eq. (3) leads to 4(|S | − 2) ≥ k|S | ≥ 4|S |, a contradiction.  We emphasize that Corollary 5 and Proposition 6 lead to infinite families of graphs whose Cartesian product satisfies

φkd (G1 G2 ) = n1 n2 . For instance, if G1 is a tree of order n1 and maximum degree ∆1 ≥ 2, G2 is a graph of order n2 and maximum degree ∆2 , and k ∈ {2 + ∆2 , . . . , ∆1 + ∆2 }, we have φkd (G1 G2 ) = φkd−∆2 (G1 )n2 = n1 n2 . In particular, if G2 is a cycle graph, then φ4d (G1 G2 ) = n1 n2 .

Another example of equality in Corollary 5 (ii) is obtained, for instance, taking the Cartesian product of the star graph St of order t + 1 and the path graph Pr of order r. In that case, for G1 = St we have δ1 = 1, n1 = t + 1 and φ0d (G1 ) = t, and, for G2 = Pr we have δ2 = 1, n2 = r and φ1d (G2 ) = r − 1. Therefore, φ0d (G1 )φ1d (G2 )+ min{n1 −φ0d (G1 ), n2 −φ1d (G2 )} = t (r − 1)+ 1. On the other hand, it is not difficult to check that, if we take all leaves belonging to the copies of St corresponding to the first r − 1 vertices of G2 and we add the vertex of degree t belonging to the last copy of St , we obtain a maximum defensive 0-alliance free set of cardinality t (r − 1) + 1 in the graph G1 G2 , that is, φ0d (G1 G2 ) = t (r − 1) + 1. See Fig. 1. This example   also shows that this bound is better than the bound obtained in Remark 3, which is t 2r + 1. In this particular case, both bounds are equal if and only if r = 2 or r = 3. Theorem 7. Let Gi = (Vi , Ei ) be a graph and let Si ⊆ Vi , i ∈ {1, 2}. If S1 × S2 is a k-daf set in G1 G2 and S2 is a defensive k′ -alliance in G2 , then S1 is a (k − k′ )-daf set in G1 . Proof. If S ⊆ S1 , then S × S2 ⊆ S1 × S2 is a k-daf set in G1 G2 . So, there exists (a, b) ∈ S × S2 such that δS ×S2 (a, b) < δS ×S2 (a, b) + k. Thus, we have

δS (a) + δS2 (b) = δS ×S2 (a, b) < δS ×S2 (a, b) + k = δS (a) + δS2 (b) + k.

(4)

As S2 is a defensive k -alliance in G2 , for every b ∈ S2 we have, δS2 (b) ≥ δS2 (b) + k . Hence, from Eq. (4), we obtain δS (a) < δS (a) + k − k′ . Therefore, S is not a defensive (k − k′ )-alliance in G1 and, as a consequence, S1 is a (k − k′ )-daf set.  ′



Taking into account that V2 is a defensive δ2 -alliance in G2 we obtain the following result. Corollary 8. Let Gi = (Vi , Ei ) be a graph, i ∈ {1, 2}. Let δ2 be the minimum degree of G2 and let S1 ⊆ V1 . If S1 × V2 is a k-daf set in G1 G2 , then S1 is a (k − δ2 )-daf set in G1 . By Theorem 4(i) and Corollary 8 we obtain the following result. Proposition 9. Let G1 be a graph of maximum degree ∆1 and let G2 be a δ2 -regular graph. For every k ∈ {δ2 − ∆1 , . . . , ∆1 + δ2 }, S1 × V2 is a k-daf set in G1 G2 if and only if S1 is a (k − δ2 )-daf set in G1 .

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Fig. 2. The graph G = (V , E ) is the Cartesian product of the cycle graph C3 by the path graph P3 where S = V \ {(1, 3), (2, 3)} is a maximum offensive 3-alliance free set.

4. Offensive k-alliance free sets in Cartesian product graphs Theorem 10. Let Gi = (Vi , Ei ) be a graph, i ∈ {1, 2}, and let S ⊂ V1 × V2 . If PVi (S ) is a k-oaf set in Gi , then S is a (k − δj )-oaf set in G1 G2 , where δj denotes the minimum degree of Gj and j ∈ {1, 2}, i ̸= j. Proof. If PV1 (S ) is a k-oaf set in G1 and A ⊆ S, then PV1 (A) ⊆ PV1 (S ) is a k-oaf set in G1 . So, there exists a ∈ ∂ PV1 (A), such that δPV (A) (a) < δPV (A) (a) + k. Let a′ ∈ PV1 (A) such that a and a′ are adjacent, and let Ya′ be the set of elements of A whose 1

1

first component is a′ . Thus, if b ∈ PV2 (Ya′ ), then (a, b) ∈ ∂ A, so we have

δA (a, b) ≤ δPV1 (A) (a) < δPV

1

(A) (a)

+ k ≤ δA (a, b) − δ(b) + k ≤ δA (a, b) + k − δ2 .

Therefore, A is not an offensive (k − δ2 )-alliance in G1 G2 . The proof of the other case is completely analogous.



From Theorem 10 we conclude that for every ki -oaf set Si in Gi , i ∈ {1, 2}, the sets S1 × V2 and V1 × S2 are (k1 − δ2 )-oaf and (k2 − δ1 )-oaf, respectively, in G1 G2 . Therefore, we obtain the following result. Corollary 11. Let Gl be a graph of order nl , maximum degree ∆l and minimum degree δl , l ∈ {1, 2}. Then, for every k ∈ {2 − δj − ∆i , . . . , ∆i − δj }, φko (G1 G2 ) ≥ nj φko+δj (Gi ), where i, j ∈ {1, 2}, i ̸= j. Example of equality in the above result is the following. If we take G1 = C4 , G2 = P3 and k2 = 2, then φ0o (C4 P3 ) = 8 = 4φ2o (P3 ). Theorem 12. Let Gi = (Vi , Ei ) be a graph of minimum degree δi and maximum degree ∆i . If Si is a ki -oaf set in Gi , i ∈ {1, 2}, then for every k ∈ {k′ , . . . , ∆1 + ∆2 }, (S1 × V2 ) ∪ (V1 × S2 ) is a k-oaf set in G1 G2 , where k′ = max{k1 − δ2 , k2 − δ1 , min{k2 + ∆1 , k1 + ∆2 }}. Proof. Let A ⊆ (S1 × V2 ) ∪ (V1 × S2 ). By Theorem 10 we deduce that, if A ⊆ S1 × V2 , then A is a (k1 − δ2 )-oaf set in G1 G2 . Analogously, if A ⊆ V1 × S2 , then A is a (k2 − δ1 )-oaf set in G1 G2 . Now we suppose A ̸⊆ S1 × V2 and A ̸⊆ V1 × S2 . Let B = A \ (S1 × V2 ). For every a ∈ PV1 (B), the set Ya , composed of the elements of B whose first component is a, satisfies that PV2 (Ya ) is a k2 -oaf set in G2 . Then, there exists b ∈ ∂ PV2 (Ya ) such that δPV (Ya ) (b) < δPV (Ya ) (b) + k2 . Also, notice that (a, b) ∈ ∂ A. Thus, 2

2

δA (a, b) ≤ δPV2 (Ya ) (b) + δ(a) < δPV

2

(Ya ) (b)

+ k2 + δ(a) ≤ δA (a, b) + k2 + ∆1 .

We conclude that A is not an offensive (k2 + ∆1 )-alliance in G1 G2 . Analogously, A is not an offensive (k1 + ∆2 )-alliance in G1 G2 . Therefore, the result follows.  Corollary 13. Let Gi be a graph of order ni , minimum degree δi and maximum degree ∆i , i ∈ {1, 2}. Let k′ = max{k1 − δ2 , k2 − δ1 , min{k2 + ∆1 , k1 + ∆2 }}, where ki ∈ {2 − ∆i , . . . , ∆i }. Then, for every k ∈ {k′ , . . . , ∆1 + ∆2 },

φko (G1 G2 ) ≥ n1 φko2 (G2 ) + n2 φko1 (G1 ) − φko1 (G1 )φko2 (G2 ). For instance, if we take G1 = C3 , G2 = P3 , k1 = 1 and k2 = 2, then φ3o (C3 P3 ) = 7 = 3φ2o (P3 ) + 3φ1o (C3 ) − φ1o (C3 )φ2o (P3 ). (see Fig. 2). 5. Powerful k-alliance free sets in Cartesian product graphs p

Since for every graph G, φk (G) ≥ max{φkd (G), φko+2 (G)}, we have that lower bounds on φkd (G) and φko+2 (G) lead to lower p bounds on φk (G). So, by the results obtained in the above sections on φkd (G1 G2 ) and φko+2 (G1 G2 ) we deduce lower bounds p on φk (G1 G2 ).

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Fig. 3. A graph G = (V , E ) where V is a powerful 2-alliance free set, although {2, 3, 4, 5, 6, 8} is a defensive 2-alliance and {3, 4, 5, 6, 7} is an offensive 4-alliance. p

We emphasize that there are graphs where φk (G) > max{φkd (G), φko+2 (G)}. For instance, the graph of Fig. 3 satisfies φ (G) = 9 while φ2d (G) = 8 and φ4o (G) = 7. p 2

Theorem 14. Let Gi = (Vi , Ei ) be a simple graph of maximum degree ∆i and minimum degree δi , i ∈ {1, 2}, and let S ⊆ V1 × V2 . Then the following assertions hold. (i) If for some i ∈ {1, 2}, PVi (S ) is a ki -paf set in Gi , then, for every k ∈ {ki + ∆j , . . . , ∆i + ∆j − 2}, S is a k-paf set in G1 G2 , where j ∈ {1, 2}, j ̸= i. (ii) If for every i ∈ {1, 2}, PVi (S ) is a ki -paf set in Gi , then, for every k ∈ {k′ , . . . , ∆1 + ∆2 − 2}, S is a k-paf set in G1 G2 , where k′ = max{k1 + k2 − 1, min{k2 − δ1 , k1 − δ2 }}. Proof. Let A ⊆ S. We suppose PVi (S ) is a ki -paf set in Gi for some i ∈ {1, 2}. Since PVi (A) ⊆ PVi (S ), it follows that PVi (A) is not a powerful ki -alliance in Gi . If PVi (A) is not a defensive ki -alliance, by analogy to the proof of Theorem 4(i), we obtain that A is not a defensive (ki + ∆j )-alliance in G1 G2 , j ̸= i. If PVi (A) is not an offensive (ki + 2)-alliance in Gi , then by analogy to the proof of Theorem 10, we obtain that A is not an offensive (ki − δj + 2)-alliance in G1 G2 , j ̸= i. Since, ki + ∆j > ki − δj , we obtain that A is not a powerful (ki + ∆j )-alliance in G1 G2 . Therefore, (i) follows. If for every l ∈ {1, 2}, PVl (S ) is a kl -paf set in Gl , then PVl (A) is not a powerful kl -alliance in Gl . Hence, we differentiate two cases. Case (1): For some l ∈ {1, 2}, PVl (A) is not a defensive kl -alliance. We suppose PV1 (A) is not a defensive k1 -alliance. Hence, there exists x ∈ PV1 (A) such that δPV (A) (x) < δPV (A) (x) + k1 . Let Ax ⊆ A be the set composed of the elements of A whose first 1

1

component is x. If PV2 (Ax ) ⊂ PV2 (S ) is not a defensive k2 -alliance, then by analogy to the proof of Theorem 4(ii), we obtain that A is not a defensive (k1 + k2 − 1)-alliance in G1 G2 . On the other hand, if PV2 (Ax ) is a defensive k2 -alliance, then it is not an offensive (k2 + 2)-alliance. Thus, there exists y ∈ ∂ PV2 (Ax ) such that δPV (Ax ) (y) < δPV (Ax ) (y) + (k2 + 2). We note that 2

(x, y) ∈ ∂ A. Hence,

2

δA (x, y) ≤ δPV1 (A) (x) + δPV2 (Ax ) (y) < δPV

1

(A) (x)

+ δPV

2

(Ax ) (y)

+ k1 + k2 + 1

≤ δA (x, y) + k1 + k2 + 1. As a consequence, A is not an offensive (k1 + k2 + 1)-alliance in G1 G2 . Thus, in this case, A is not a powerful (k1 + k2 − 1)alliance in G1 G2 . Case (2): For every i ∈ {1, 2}, PVi (A) is not an offensive (ki + 2)-alliance in Gi . In this case, as we have shown in the proof of (i), A is not an offensive (ki − δj + 2)-alliance in G1 G2 , j ∈ {1, 2}, j ̸= i. As a consequence, for k = max{k1 + k2 − 1, k1 − δ2 , k2 − δ1 }, A is not a powerful k-alliance in G1 G2 . Hence, S is a k-paf set in G1 G2 . Therefore, (ii) follows.  Corollary 15. Let Gl be a graph of order nl , maximum degree ∆l and minimum degree δl , l ∈ {1, 2}. Let kl ∈ {1 −δl , . . . , ∆l − 2}. Then the following assertions hold. (i) For every k ∈ {∆j − ∆i , . . . , ∆i + ∆j − 2}, (i, j ∈ {1, 2}, i ̸= j)

φkp (G1 G2 ) ≥ nj φkp−∆j (Gi ). (ii) For every k ∈ {k1 + k2 − 1, . . . , ∆1 + ∆2 − 2},

φkp (G1 G2 ) ≥ φkp (G1 )φkp (G2 ) + min{n1 − φkp (G1 ), n2 − φkp (G2 )}. 1

2

1

2

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Proof. By Theorem 14(i) we conclude that for every ki -paf set Si in Gi , i ∈ {1, 2}, the sets S1 × V2 and V1 × S2 are, respectively, (k1 + ∆2 )-paf and (k2 + ∆1 )-paf in G1 G2 . Therefore, (i) follows. In order to prove (ii), let V1 = {u1 , u2 , . . . , un1 } and V2 = {v1 , v2 , . . . , vn2 }. Let Si be a ki -paf set of maximum cardinality in Gi , i ∈ {1, 2}. We suppose S1 = {u1 , . . . , ur } and S2 = {v1 , . . . , vs }. By Theorem 14(ii) we deduce that, for k ≥ k1 + k2 − 1, S1 × S2 is a k-paf set in G1 G2 . Now let X = {(ur +i , vs+i ), i = 1, . . . , t }, where t = min{n1 − r , n2 − s} and let S = X ∪ (S1 × S2 ). Since, for every x ∈ X , δS (x) = 0 and ki > −δi , i ∈ {1, 2}, we obtain that for every A ⊆ S, such that A ∩ X ̸= ∅, A is not a defensive (k1 + k2 − 1)-alliance in G1 G2 . Hence, S is a k-paf set for k ≥ k1 + k2 − 1. As a consequence, φkp (G1 G2 ) ≥ |S | = φkp (G1 )φkp (G2 ) + min{n1 − φkp (G1 ), n2 − φkp (G2 )}.  1

2

1

2

p

If G1 = Cn1 is the cycle graph of order n1 and G2 is the graph in Fig. 3, then, by Corollary 15(i), we deduce φ4 (G1 G2 ) = p

p

n1 n2 , that is, φ4 (G1 G2 ) ≥ n1 φ2 (G2 ) = n1 n2 . Moreover, if G1 = Tn1 is a tree of order n1 and maximum degree ∆1 ≥ 4 and p p p G2 is the graph in Fig. 3, then φ2 (G1 ) = n1 and φ2 (G2 ) = n2 = 9. Therefore, by Corollary 15(ii) we deduce φ3 (G1 G2 ) = 9n1 . 6. Conclusions This article is a contribution to the study of alliances in graphs. Particularly, we have dealt with defensive (offensive, powerful) k-alliance free sets in Cartesian product graphs. We have shown several relationships between defensive (offensive, powerful) k-alliance free sets in Cartesian product graphs and defensive (offensive, powerful) k-alliance free sets in the factor graphs. Our principal contributions are summarized below. Let Gi = (Vi , Ei ) be a graph of maximum degree ∆i and minimum degree δi , i ∈ {1, 2}:

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