global powerful r-alliances and total k-domination in graphs
TRANSCRIPT
Global powerful r-alliances and total
k-domination in graphs
H. Fernau∗1, J. A. Rodrıguez-Velazquez †2 and J. M. Sigarreta‡31FB 4-Abteilung Informatik Universitat Trier, 54286 Trier, Germany.
2Departament d’Enginyeria Informatica i Matematiques
Universitat Rovira i Virgili, Av. Paısos Catalans 26, 43007 Tarragona, Spain.3 Faculty of Mathematics, Autonomous University of Guerrero
Carlos E. Adame 5, Col. La Garita, Acapulco, Guerrero, Mexico
December 13, 2011
Abstract
Let G = (V,E) denote a simple graph of order n and size m.For a non-empty subset S ⊂ V , and a vertex v ∈ V , we denote byNS(v) the set of neighbors v has in S. We denote the degree of v inS by degS(v) = |NS(v)|. The boundary of a set S ⊆ V is defined as
∂S =⋃v∈S
NS(v). S denotes the complement of S, i.e., S = V \ S.
A set D ⊆ V is called dominating if, for each v ∈ D, ND(v) 6= ∅.In this paper, we consider several variants of dominating sets.
Consider the following condition (∗): degS(v) ≥ deg S(v) + r,which states that a vertex v has at least r more neighbors in S thanit has in S. A non-empty set S ⊆ V that satisfies Condition (∗)for every vertex v ∈ S is called a defensive r-alliance; if S 6= ∅satisfies Condition (∗) for every vertex v ∈ ∂S, then S is calledan offensive r-alliance. A set S ⊂ V is a powerful r-alliance in Gif S is both a defensive r-alliance and an offensive (r + 2)-alliancein G. An alliance is called global if it is also a dominating set. Theglobal powerful r-alliance number, denoted by γ∗r (G), is defined as theminimum cardinality of a global powerful r-alliance in G. We showthat the natural decision problem associated to computing γ∗r (G) isNP-complete and we obtain tight bounds for this domination-relatedparameter.
∗e-mail:[email protected]†e-mail:[email protected]‡e-mail:[email protected]
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A set S ⊂ V is a total k-dominating set if degS(v) ≥ k, ∀v ∈ V .The total k-domination number γkt(G) is the minimum cardinalityof a total k-dominating set. We show that the problem of computingγkt(G) is NP-complete and we obtain a tight bound for this param-eter, as well.
Finally, we exhibit close relationships between total k-dominatingsets and global (defensive / offensive / powerful) r-alliances, with rand k tightly coupled.
1 Introduction
Since (defensive, offensive and powerful) alliances were first introduced byKristiansen, Hedetniemi and Hedetniemi [20], several authors have studiedtheir mathematical properties [1, 2, 4, 7, 15, 23, 24, 25, 26, 27, 30, 33, 35] aswell as the complexity of computing the minimum cardinality of alliances [3,9, 16, 17]. The minimum cardinality of a defensive (respectively, offensiveor powerful) alliance in a graph G is called the defensive (respectively,offensive or powerful) alliance number of G . The mathematical propertiesof defensive alliances were first studied in [20] where several bounds onthe defensive alliance number were given. The particular case of global(strong) defensive alliances was investigated in [15] where several bounds onthe global (strong) defensive alliance number were obtained. The powerfulalliances were introduced and studied in [1, 2] as alliances that are bothdefensive and offensive. In [24] there were obtained several tight boundson the defensive (offensive and powerful) alliance number. In particular,the relationships of the alliance numbers of a graph to several other graphparameters were investigated, e.g., to its algebraic connectivity, its spectralradius, and its Laplacian spectral radius. Alliances that are also dominatingsets are called global alliances. The study of global defensive (offensive andpowerful) alliances in planar graphs was initiated in [25] and the study ofdefensive alliances in the line graph of a simple graph was initiated in [33].The particular case of global alliances in trees has been investigated in [4].For many properties of offensive alliances, readers are referred to [7, 23, 34].
A generalization of (defensive and offensive) alliances called r-allianceswas presented by Shafique and Dutton [28, 29, 30]. There, the study of r-alliance free sets and r-alliance cover sets was initiated. The aim of thiswork is to study mathematical properties of powerful r-alliances and totalk-dominating sets. We begin by stating the terminology used. Throughoutthis article, G = (V,E) denotes a simple graph of order |V | = n andsize |E| = m. We denote two adjacent vertices u and v by u ∼ v. Fora nonempty set X ⊆ V , and a vertex v ∈ V , NX(v) denotes the set ofneighbors v has in X: NX(v) := {u ∈ X : u ∼ v}, and the degree of vin X will be denoted by degX(v) = |NX(v)|. We denote the degree of a
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vertex v ∈ V by deg(v). The maximum and minimum degrees in G will bedenoted by ∆ and δ, respectively. The subgraph induced by S ⊂ V will bedenoted by G [S], i.e.,
G [S] = (S, {uv ∈ E : u, v ∈ S})
and the complement of the set S in V will be denoted by S.A nonempty set S ⊂ V is called dominating if, for each v ∈ S, NS(v) 6=
∅. A dominating set for G of minimal cardinality is also termed a minimumdominating set, and its cardinality is denoted by γ(G). In the next sections,we will employ the following derived decision problem that is well-knownto be NP-complete:
dominating set (DS)INSTANCE: A graph G and a bound ` ∈ NQUESTION: Is γ(G) ≤ `?
It is known that the dominating set problem, restricted to graphsof certain fixed minimum degree, is still NP-hard. For this fact and notionsrelated to NP-completeness, we refer the reader to [12].
A nonempty set S ⊆ V is a defensive r-alliance in G = (V,E), withinteger r in the range −∆ ≤ r ≤ ∆, if
degS(v) ≥ deg S(v) + r, for every v ∈ S. (1)
If r ≥ 0, then this condition can be only satisfied for S where the minimumdegree in G [S] is at least r. Hence, in graphs G with ∆(G) < r, there isno defensive r-alliance.
A vertex v ∈ S is said to be r-satisfied with respect to S if (1) holds.Notice that (1) is equivalent to
deg(v) ≥ 2deg S(v) + r, for every v ∈ S. (2)
A defensive (−1)-alliance is a defensive alliance and a defensive 0-alliance is a strong defensive alliance as defined in [20]. A defensive 0-alliance is also known as a cohesive set [31].
A defensive r-alliance S is called global if it is a dominating set. Theglobal defensive r-alliance number γdr (G) is the minimum cardinality of anyglobal defensive r-alliance in G .
The boundary of a set S ⊆ V is defined as ∂S =⋃v∈S
NS(v). A non-
empty set of vertices S ⊆ V is called offensive r-alliance in G if
degS(v) ≥ deg S(v) + r, for every v ∈ ∂S (3)
where −∆+2 < r ≤ ∆. In particular, an offensive 1-alliance is an “offensivealliance”, and an offensive 2-alliance is a “strong offensive alliance” (asdefined in [20]).
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A non-empty set of vertices S ⊆ V is a global offensive r-alliance if
degS(v) ≥ deg S(v) + r, for every v ∈ S. (4)
The global offensive r-alliance number, denoted by γor (G), is defined as theminimum cardinality of a global offensive r-alliance in G .
An alliance is called powerful if it is both defensive and offensive [20].Hence, a global powerful alliance is a global powerful (−1)-alliance, i.e.,defensive (−1)-alliance and offensive 1-alliance, and a global strong power-ful alliance is a global powerful 0-alliance, i.e., (0)-defensive alliance andoffensive 2-alliance. In general, a set S ⊂ V is a powerful r-alliance in Gif S is both defensive r-alliance and offensive (r + 2)-alliance in G . Forpowerful r-alliances the integer r is in the range 1−∆ to ∆−2. The globalpowerful r-alliance number, denoted by γ∗r (G), is defined as the minimumcardinality of a global powerful r-alliance in G .
The global powerful alliance number γ∗−1 is known for some familiesof graphs such as the complete graph Kn, the path graph Pn, the completebipartite graph Kp,s, the cycle graph Cn and the wheel graph of order n+1,Wn.
Remark 1. [32]
• γ∗−1(Kn) =⌈n2
⌉.
• γ∗−1(Pn) =⌊
2n3
⌋.
• γ∗−1(Cn) =⌈
2n3
⌉.
• p ≤ s, γ∗−1(Kp,s) = min{⌈
p+12
⌉+⌈s+1
2
⌉, p+
⌊s2
⌋}.
• γ∗−1(Wn) =⌈n+1
2
⌉.
Fink and Jacobson [11] generalized the concept of domination andintroduced the so called k-dominating sets. A set S ⊂ V is a k-dominatingset if
degS(v) ≥ k, for every v ∈ S. (5)
Other generalization of the concept of domination is the concept of totaldomination, introduced by Cockayne, Dawes and Hedetniemi in [5]: a setS ⊂ V is a total dominating set if degS(v) ≥ 1, ∀v ∈ V . The total domina-tion number γ
t(G) is the minimum cardinality of a total dominating set.
So, a set S ⊂ V is a total k-dominating set if
degS(v) ≥ k, for every v ∈ V. (6)
Therefore, every total k-dominating set is a k-dominating set, but the con-verse is not true. The total k-domination number γ
kt(G) is the minimum
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cardinality of a total k-dominating set. Notice that the concept of total2-domination is different from the concept of double domination introducedby Harary and Haynes in [14]. A set S ⊂ V is a double dominating set ifdegS(v) ≥ 2, ∀v ∈ S and degS(v) ≥ 1, ∀v ∈ S. A fortiori every doubledominating set is a total 1-dominating set. Moreover, the concept of totalk-dominating set coincides with the concept of k-tuple total dominating setintroduced by Henning and Kazemi in [18].
The total 2-domination number γ2t is easy to compute in the caseof some families of graphs such as the complete graph Kn, the completebipartite graph Kp,s and the cycle graph Cn.
Remark 2.
• γ2t
(Kn) = 3.
• γ2t(Cn) = n.
• 2 ≤ p ≤ s, γ2t(Kp,s) = 4.
2 The global powerful r-alliance number
In this section, we address the complexity of the following problem, alsocalled r-GPA for short: Given a graph G and a bound `; determine ifγ∗r (G) ≤ `. More precisely, for any fixed r, we consider the followingdecision problem:
global powerful r-alliance (r-GPA)INSTANCE: A graph G and a bound ` ∈ NQUESTION: Is γ∗r (G) ≤ `?
Theorem 3. For all fixed r, the problem r-GPA is NP-complete.
We are using a construction based on ideas presented by Cami etal. [3] for the special case r = −1, although our construction is differentconcerning some details.
Proof. (A) Membership in NP is quite clear for each r: a nondeterministicTuring machine first guesses at most ` vertices and then verifies that thisvertex set forms indeed a global powerful r-alliance. The latter propertycan be tested in polynomial time, on a deterministic Turing machine.
(B) Now, fix an r ≥ −1. We consider graphs G = (V,E) of minimumdegree four as instances of dominating set, together with a bound k onthe dominating set size. We show how to construct an r-GPA instanceG ′ = (V ′, E′) together with a bound k′, such that G has a dominating set
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of size at most k if and only if G ′ has a global powerful r-alliance of sizeat most k′.
The aforementioned graph G ′ has G as a subgraph. To each v ∈ V ,NV ′(v) contains degV (v) + r new neighbors, which are collected in thevertex set A(v). G ′[A(v)] forms a clique. Moreover, to v we associate anindependent set B(v) of size degV (v)−2. Every x ∈ B(v) has r+2 neighborsin A(v) (and no other neighbors). Notice that, due to r ≥ −1, the set ofneighbors of each such x is non-empty. We assume that B(v) is “evenlydistributed” among the A(v), meaning that the number b(x) and b(y) ofB(v)-neighbors of arbitrary x ∈ A(v) and y ∈ A(v) obey: |b(x)− b(y)| ≤ 1.
Hence, for the average b = (r+2)(degV (v)−2)degV (v)+r we find |b − b(y)| ≤ 1 for all
y ∈ A(v). We derive an estimate for b in the following:
b =rdegV (v)− 2r − 4
degV (v) + r=
(degV (v) + r)degV (v)− deg2V (v) + degV (v)− 2r − 4
degV (v) + r
= degV (v) +−(degV (v)− 1)2 − 2r − 3
degV (v) + r.
Observe that degV (v) ≥ 4 implies that (degV (v) − 1)2 ≥ 2degV (v), whichshows b < degV (v)− 2. Hence, b(x) ≤ degV (v)− 2 for each x ∈ A(v).
We first show that if D ⊆ V is a dominating set for G , then S =D ∪
⋃v∈V A(v) is a solution to the r-GPA instance G ′.
Consider some u ∈ B(v). By construction, u ∈ ∂S, and u has r + 2neighbors in A(v) and hence in S, but no neighbors in S.Now, discuss the case when u ∈ A(v). On the one hand,
|NV ′(u) ∩ S| ≥ |A(v)| − 1 = degV (v) + r − 1,
since A(v) ⊆ S. On the other hand, |NV ′(u) ∩ S| ≤ (degV (v) − 2) + 1.Hence, |NV ′(u) ∩ S| − |NV ′(u) ∩ S| ≥ r, as required.Finally, consider some v ∈ V . There are two possible cases: (1) v ∈ D, i.e.,v ∈ S. Then, |NV ′(v) ∩ S| ≥ |A(v)| = degV (v) + r, while |NV ′(v) ∩ S| ≤degV (v), so that |NV ′(v)∩S|−|NV ′(v)∩S| ≥ r, as required by the defensivealliance condition. (2) v /∈ D, i.e., v ∈ S. Then, |NV ′ ∩ S| = |A(v) + (D ∩NV (v))| = degV (v) + r+ |D∩NV (v)|. Similarly, |NV ′ ∩ S| = |S∩NV (v)| =degV (v) − |D ∩ NV (v)|. Therefore, since D is a dominating set, we have|NV ′ ∩ S| − |NV ′ ∩ S| = r + 2|D ∩ NV (v)| ≥ r + 2 as required. Noticethat the converse is also true: if S is constructed as described, namely asthe union of A(v)-vertices and some vertices from V , then there must bea vertex from V ∩ S in the neighborhood of any vertex from V ∩ S, sinceotherwise v ∈ ∂S (since A(v) ⊆ S) would not have more than degV (v) + rneighbors in S as required by the offensive alliance condition; hence, S ∩Vforms a dominating set in G .
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Now, consider a set S that forms a minimal powerful r-alliance in G ′
of size at most k+∑v∈V (degV (v)+r) = k+2|E|+r|V |. If, for some v ∈ V ,
B(v)∩S 6= ∅, then at least r+1 of the r+2 neighbors of x ∈ B(v)∩S mustbe in S, too. Since x is dominating only its neighbors in A(v), we cannotincrease the size of S by deleting x from S and putting all neighbors fromx into S, instead, and we would still find a powerful alliance. Continuingthis way, we end up with some powerful alliance S satisfying A(v) ⊆ Sand B(v) ∩ S = ∅ for all v ∈ V . Our computations above show that suchalliances have the property that S ∩ V forms a dominating set of size atmost k in G , as required.
(C) Observe that, for r ≤ −3, on cubic graphs, each vertex set S is a globaldefensive r-alliance if and only if S is a dominating set. Moreover, forr ≤ −1, on cubic graphs, each vertex set S is a global offensive r-allianceif and only if S is a dominating set. Since the Dominating Set problemis known to be NP-hard on cubic graphs, cf. [19], the claim easily followsfor r ≤ −3.
(D) The only remaining case is r = −2. In that case, we can provide similarobservations as in (C) concerning the problem of total dominating seton cubic graphs, instead. This (restricted) problem was shown to be NP-complete in [21, Theorem 4.3.6].
The following result leads to several bounds on γ∗r (G).
Theorem 4. Let G be a graph of order n, size m, minimum degree δ andmaximum degree ∆. If S is a global powerful r-alliance in G, then the sizeof G [S] is bounded below by 1
4 (m+ |S|(r − 1)) + 18n(r + 2).
Proof. If S ⊂ V is a global offensive (r + 2)-alliance, then∑v∈S
degS(v) ≥∑v∈S
deg S(v) + (n− |S|)(r + 2). (7)
Hence, as∑v∈S
deg S(v) =∑v∈S
degS(v),
∑v∈S
degS(v) ≥
2m−∑v∈S
degS(v)− 2∑v∈S
degS(v)
+(n−|S|)(r+2). (8)
Thus,
3∑v∈S
degS(v) +∑v∈S
degS(v) ≥ 2m+ (n− |S|)(r + 2). (9)
On the other hand, if S is a global defensive r-alliance in G ,∑v∈S
degS(v) ≥∑v∈S
deg S(v) + r|S|. (10)
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Therefore, by (9) and (10) we have
4∑v∈S
degS(v) ≥ 2m+ n(r + 2) + 2|S|(r − 1). (11)
As the size of G [S] is 12
∑v∈S
degS(v), the result follows.
Corollary 5. For any graph G of order n and size m,
γ∗r (G) ≥
⌈√8m+ 4n(r + 2) + (r + 1)2 + r + 1
4
⌉.
Proof. If S ⊂ V is a global powerful r-alliance, by Theorem 4 and |S|(|S|−1) ≥
∑v∈S
degS(v), we obtain 4|S|(|S| − 1) ≥ 2m + n(r + 2) + 2|S|(r − 1).
Hence, the lower bound follows.
As we will show in Remark 7, the above bound is attained for thefamily of complete graphs of order n.
Theorem 6. Let G be a graph of order n and minimum degree δ ≥ 1. Let
α = min{δ,∆− 2}. If r ∈ {1− δ, ..., α}, then γ∗r (G) ≤ n−⌊δ − r
2
⌋.
Proof. Let x be a vertex of minimum degree in G = (V,E) and let Y ⊂NV (x) such that |Y | =
⌈δ+r
2
⌉. Let X = {x} ∪ NV (x) − Y . As r + δ ≥ 1,
Y 6= ∅, so x has at least one neighbor in X. Moreover, if u ∈ X − {x},then u has at least one neighbour in X, because otherwise deg(u) < δ, acontradiction. Hence, X is a dominating set.
On the one hand, for all v ∈ X we have degX(v) ≥⌈δ+r
2
⌉− 1 ≥
δ+ r−⌈δ+r
2
⌉− 1 ≥ degX(v) + r− 2. Thus, X is a global defensive (r− 2)-
alliance in G .On the other hand, since degX(x) =
⌈δ+r
2
⌉≥⌊δ+r
2
⌋= δ−
⌈δ+r
2
⌉+r =
degX(x) + r, we have degX(u) ≥ degX(x) ≥ degX(x) + r ≥ degX(u) +r, ∀u ∈ X. Therefore, X is a global offensive r-alliance in G . As aconsequence, X is a global powerful (r−2)-alliance in G . Hence, γ∗r−2(G) ≤n− 1− δ +
⌈δ+r
2
⌉. Thus, the bound follows:
γ∗r (G) ≤ n−1−δ+⌈δ + r + 2
2
⌉= n+
⌈(−2− 2δ) + δ + r + 2
2
⌉= n−
⌊δ − r
2
⌋.
From Theorem 6 and Corollary 5 we deduce the exact value of γ∗r (G)for the complete graph G = Kn.
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Remark 7. γ∗r (Kn) =
⌈n+ r + 1
2
⌉.
We now relate the graph parameter γ∗r with another well-known pa-rameter, the spectral radius, which is defined as the maximum eigenvalueof the adjacency matrix of the graph.
Theorem 8. For any graph G of order n, size m, and spectral radius λ,
γ∗r (G) ≥⌈
2m+ (r + 2)n
4λ− 2r + 2
⌉.
Proof. Let S be a global powerful r-alliance in G . Let A be the adjacencymatrix of G . Then
λ ≥ 〈Aw,w〉〈w,w〉
, for every w ∈ Rn \ {0}, (12)
where 〈, 〉 is the standard scalar product in the Euclidean space Rn. Thus,taking w ∈ Rn defined as
wi =
{1 if vi ∈ S;0 otherwise.
we haveλ|S| ≥
∑v∈S
degS(v). (13)
Therefore, by (13) and Theorem 4 we obtain 4λ|S| ≥ 2m + n(r + 2) +2|S|(r − 1). So, the result follows.
The above bound is tight. For the left-hand side graph of Figure 1,Theorem 8 leads to γ∗−1(G1) ≥ d 2·9+10
4√
6+4e = 3 and, for the right-hand side
graph of Figure 1, Theorem 8 leads to γ∗0 (G2) ≥ d 2·9+2·64(1+
√5)+2e = 3.
Figure 1: The spectral radius of the left-hand side graph G1 is λ =√
6 andthe spectral radius of the right-hand side graph G2 is λ = 1 +
√5.
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In the particular case of cubic graphs1 we have:
n
4≤ γ(G) = γ∗−3(G) ≤ γ∗−2(G) = γ∗−1(G) ≤ γ∗0(G) = γ∗1(G).
So, in this case we only study γ∗−1(G) and γ∗0 (G). As a consequence ofCorollary 5 we obtain that for any cubic graph of order n,
γ∗−1(G) ≥√n. (14)
Theorem 9. For any cubic graph G of order n, γ∗0 (G) ≥ 3n4 .
Proof. As G = (V,E) is a cubic graph we have that if S ⊂ V is a globalpowerful 0-alliance in G , then S is an independent set. Therefore,
3(n− |S|) ≤ |S|. (15)
By solving (15) for |S| we obtain the result.
In the case of G = C3 ×K2 Theorem 9 leads to γ∗0(G) ≥ 5. Thus, thebound is tight.
2.1 Powerful r-alliances in planar graphs
Theorem 10. Let G be a graph of order n and size m. Let S be a globalpowerful r-alliance in G such that G [S] is a planar graph.
(i) If n > 2(2− r), then |S| ≥⌈
2(m+24)+(r+2)n2(13−r)
⌉.
(ii) If n > 2(2−r) and G [S] is a triangle free graph, then |S| ≥⌈
2(m+16)+(r+2)n2(9−r)
⌉.
Proof.
(i) If |S| ≤ 2, for all v ∈ S we have deg S(v) ≤ 1− r. Thus, n ≤ 2(2− r).It contradicts that n > 2(2− r). Thus |S| > 2.
As G [S] is planar and |S| > 2, the size of G [S] is bounded by
1
2
∑v∈S
degS(v) ≤ 3(|S| − 2). (16)
Hence, by Theorem 4 and (16) we obtain 14 (m+|S|(r−1))+ 1
8n(r+2) ≤3(|S| − 2), so the result follows.
1A cubic graph is a 3-regular graph, i.e., each vertex has degree three.
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(ii) If G [S] is a triangle free graph, then the size of G [S] is bounded by
1
2
∑v∈S
degS(v) ≤ 2(|S| − 2). (17)
Moreover, by Theorem 4 and (17) we have 14 (m+ |S|(r−1))+ 1
8n(r+2) ≤ 2(|S| − 2), so the result follows.
As a direct consequence of above theorem we obtain the followingresult.
Corollary 11. Let G be a planar graph of order n and size m.
(i) if n > 2(2− r), then γ∗r (G) ≥⌈
2(m+24)+(r+2)n2(13−r)
⌉.
(ii) If n > 2(2−r) and G is a triangle free graph, then γ∗r (G) ≥⌈
2(m+16)+(r+2)n2(9−r)
⌉.
In the case of the right hand side graph of Figure 2, the set S ={1, 2, 4} is a global powerful r-alliance of minimum cardinality for r = −1and 0, and (i) in Corollary 11 leads to γ∗r (G) ≥ 3. Moreover, the setS = {1, 2, 3, 4} is a global powerful 1-alliance of minimum cardinality and(i) leads to γ∗r (G) ≥ 4. On the other hand, if G = Q3, (ii) in Corollary11 leads to the exact value of γ∗r in the following cases: 2 ≤ γ∗−3(Q3) and4 ≤ γ∗−2(Q3) = γ∗−1(Q3).
Figure 2: Two planar graphs with γ∗r (G) = 3 for r ∈ {−1, 0}.
Theorem 12. Let G be a graph of order n. Let S be a global powerful r-alliance in G such that the subgraph G [S] is planar connected with f faces.Then,
|S| ≥⌈
2(m− 4f + 8) + n(r + 2)
2(5− r)
⌉.
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Proof. By Euler’s formula we have∑v∈S
degS(v) = 2(|S|+ f − 2). (18)
Thus, by Theorem 4 and (18) we obtain |S|+ f − 2 ≥ 14 (m+ |S|(r− 1)) +
18n(r + 2), and the result immediately follows.
In the case of the left-hand side graph of Figure 2, the set S = {2, 4, 6}is a global powerful r-alliance of minimum cardinality for r ∈ {−1, 0} andG [S] is planar with two faces. In this case, Theorem 12 leads to γ∗r (G) ≥ 3.
Theorem 13. Let T be a tree of order n. Let S be a global powerfulr-alliance in T such that the subgraph G [S] has c connected components.Then,
|S| ≥⌈n(r + 4) + 8c− 2
2(5− r)
⌉.
Proof. As the subgraph G [S] is a forest with c connected components,∑v∈S
degS(v) = 2(|S| − c). (19)
From Theorem 4 and (19) we have |S| − c ≥ 14 (m+ |S|(r− 1)) + 1
8n(r+ 2)Hence, the bound of |S| follows .
If S is a global strong powerful alliance in T and the subgraph inducedby S has c connected components, then c ≥ 2 (see Corollary 15 of [25]).Therefore, as a consequence of Theorem 13 we obtain that for any tree Tof order n,
γ∗−1(T ) ≥⌈n+ 2
4
⌉and γ∗0 (T ) ≥
⌈2n+ 7
5
⌉.
The above bounds were obtained in [25].
3 The total k-domination number
Given a graph G , a k-total dominating set is a set D of vertices satisfyingdegD(v) ≥ k for all v ∈ V . We consider the following related decidabilityproblem total k-dominating set for each fixed integer k ≥ 1:
total k-dominating set (k-TDS)INSTANCE: A graph G and a bound ` ∈ NQUESTION: Is there a k-dominating set D with |D| ≤ ` ?
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The smallest ` such that G together with ` forms a YES-instance ofk-TDS is denoted as γ
kt(G); this parameter is called the total k-domination
number of G . If k = 1, we write γt(G) for simplicity, according to the usein the literature.
Theorem 14. For all k ≥ 1, k-TDS is NP-complete.
Proof. This result is known for k = 1, where the problem becomes theclassical total dominating set problem.
Membership in NP is quite clear: a nondeterministic Turing machinehas to guess an `-element vertex set D and then verify that, for all v ∈ V ,degD(v) ≥ k.
We still have to show NP-hardness for k ≥ 2. We show that 1-TDScan be solved in polynomial time if k-TDS can be solved in polynomialtime. Consider a graph G = (V,E) and a parameter ` as an instance of 1-TDS. If G is trivial or if no feasible solution exists, we answer accordingly.Hence, from now on γt(G) > 1 and δ(G) ≥ 1.
Now, arbitrarily pick some v ∈ V ; for all w ∈ NV (v), we construct thegraph G ′v,w = (V ′, E′) as follows:
V ′ = V ∪ {1, . . . , k}, where V ∩ {1, . . . , k} = ∅,
E′ = E ∪ {{x,m} | x ∈ V, 2 ≤ m ≤ k}}∪ {{n,m} | 1 ≤ n < m ≤ k}} ∪ {{w, 1}}.
We claim that
γt(G) = minw∈NV (v)
(γkt
(G ′v,w)− (k − 1)).
Namely, since deg(1) = k, all neighbours of 1 must be in any totalk-dominating set, in particular, all “new” vertices 2, . . . , k plus one vertexfrom V .
Hence, a total k-dominating set D′ ⊆ V ′ induces a total 1-dominatingset on G as follows: If 1 /∈ D′, then D′ ∩ V is a total 1-dominating set forG . Now discuss 1 ∈ D′. Since γt(G) > 1, the only purpose of 1 within D′
is to dominate w. Since NV (w) 6= ∅, we distinguish two further cases: IfNV (w) ∩D′ 6= ∅, then D′ \ {1} is a total k-dominating set for G ′v,w, as inparticular any vertex from 2, . . . , k is dominated from (k− 1) vertices from2, . . . , k and by some vertices from NV (w) ∩ D′, so we can proceed as inthe previous case. Otherwise, D = (D′ \ {1})∪ {v} is a total k-dominatingset, which again leads us to the case 1 /∈ D′ considered before.
Since one vertex among the w ∈ NV (v) must be in any feasible solutionfor 1-TDS for G and since γt(G) > 1, our algorithm (based on determiningγ
kt(G ′v,w)) would find γt(G) in polynomial time, contradicting the known
NP-hardness of 1-TDS.
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Due to the simplicity of our construction, we can easily inherit W[2]-hardness and inapproximability results from what is known about 1-TD.The according definitions for the hardness notions of parameterized com-plexity, in particular the notion of W[2]-completeness, can be found in [6].In short, W[2]-hardness of a parameterized problem means that it is notbelieved that a parameterized algorithm showing membership in the classFPT can be found, which would be, in our case, having a running time ofO(f(`)p(m)), where ` is the integer parameter made explicit in the nextstatement, f is some arbitrary function, m is the size of the input graph,and p is some polynomial.
More precisely, we can state:
Corollary 15. For each k, k-TDS, parameterized with an upperbound `on the solution size, is W[2]-complete.
Proof. Hardness follows from the results on 1-TDS as stated in [6]. Mem-bership in W[2] can be easiest seen by a multitape Turing machine con-struction, as displayed in [8] for a related domination-type problem.
When talking about approximability, we mean the related minimiza-tion problem minimum k-TDS, so the task is to find, given a graph G =(V,E), a total k-dominating set D whose size can be upper-bounded bya function f depending on n = |V | times γkt(G), which is the size of anoptimum solution.
Corollary 16. For each k, there is a constant ck > 0 such that mini-mum k-TDS is not approximable within factor ck × n log(n) unless NP ⊂DTIME(nlog log(n)), where n denotes the number of vertices of the graph.
This follows directly from the corresponding fact for 1-TDS due toour construction, where this hardness assertion follows from [22] as for theclassical minimum dominating set problem, since both are deduced fromthe proven approximation hardness for the minimum set cover problem.
Due to Theorem 14 and its corollaries, we can conclude that we cannotautomate the process of obtaining nontrivial bounds for γkt(G). Notice thatif S ⊂ V is a total k-dominating set of minimum cardinality γ
kt(G), then
S − {v} is a total (k − 1)-dominating set, for all v ∈ S. Thus, if G has atotal k-dominating set, k ≥ 2, then
γkt
(G) ≥ γ(k−1)t
(G) + 1. (20)
Theorem 17. For any graph G of maximum degree ∆ and order n,
γkt
(G) ≥⌈kn
∆
⌉.
14
Proof. If S ⊂ V is a total k-dominating set in G ,
(n− |S|)k ≤∑v∈S
degS(v) =∑v∈S
deg S(v) ≤∑v∈S
(deg(v)− k) ≤ (∆− k)|S|.
The above bound is attained, for instance, in the case of the 3-cubegraph G = Q3. For k = 2 we have γ
2t(Q3) = 6.
Theorem 18. For any simple graph of order n and size m,
γkt
(G) ≥⌈
2(kn−m)
k
⌉.
Proof. For S ⊂ V we have
2m ≥ 2∑v∈S
degS(v) +∑v∈S
degS(v). (21)
Moreover, if S is a total k-dominating set, then∑v∈S
degS(v) ≥ k(n− |S|) (22)
and ∑v∈S
degS(v) ≥ k|S|. (23)
Thus, by (21), (22) and (23) we deduce the result.
As we will see after Corollary 19, the above bound is tight.
3.1 Total k-domination in planar graphs
As the size of a planar graph is bounded by m ≤ 3(n − 2) and the size oftriangle free graph is bounded by m ≤ 2(n− 2), by Theorem 18 we obtainthe following result.
Corollary 19. For any planar graph of order n,
γkt(G) ≥ 2[n(k − 3) + 6]
k.
Moreover, for any planar triangle-free graph of order n,
γkt(G) ≥ 2[n(k − 2) + 4]
k.
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Figure 3: The set S = {2, 4, 5, 6} is a total 3-dominating set in the righthand side graph. In the case of the left-hand side graph, each set of fourvertices composing a cycle is a total 2-dominating set.
The above bounds are tight. For instance, in the case of the right-hand side graph of Figure 3, Corollary 19 leads to γ3t(G) ≥ 4. Moreover,in the case of the left-hand side graph of Figure 3, Corollary 19 leads toγ2t(G) ≥ 4.
Theorem 20. For any planar graph of order n,
γkt(G) ≥⌈n(k − 2) + 16
6
⌉.
Moreover, for any planar triangle-free graph of order n,
γkt
(G) ≥⌈n(k − 2) + 12
4
⌉.
Proof. Let G = (V,E) be a planar graph and let S ⊂ V . Let G ′ = (V,E′)be the subgraph of G whose edge set E′ is the set of all edges in G withone endpoint in S and one endpoint in S. As G ′ is a planar triangle-freegraph,
2(n− 2) ≥ |E′| =∑v∈S
degS(v). (24)
If S ⊂ V is a total k-dominating set, by (22) we have
2(n− 2) ≥ k(n− |S|). (25)
Moreover, the subgraph induced by S, G [S] = (S,E′′) is planar, so
6(|S| − 2) ≥ 2|E′′| =∑v∈S
degS(v) ≥ k|S|. (26)
Notice that (26) and (25) lead to the first bound.
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Moreover, if G is a triangle-free graph,
4(|S| − 2) ≥ 2|E′′| =∑v∈S
degS(v) ≥ k|S|. (27)
Thus, by (25) and (27) the secund bound follows.
For k = 2 the first bound is attained in the case of the right hand sidegraph of Figure 1. The second bound is attained for k = 3 in the case ofthe right hand side graph of Figure 3.
4 Total k-domination and powerful r-alliances
Theorem 21.
1. Each total k-dominating set is a global defensive (offensive) r-alliance,where −∆ < r ≤ 2k −∆.
2. Each global powerful r-alliance, r ≥ 1, is a total r-dominating set.
Proof.
1. If S ⊂ V is a total k-dominating set in G and r ≤ 2k −∆, then
degS(v) ≥ k ≥ r + ∆− k ≥ r + deg(v)− k ≥ r + deg S(v), ∀v ∈ V.
Namely, v as at least k neighbours in S (being total k-dominating)and therefore at most k + deg S(v) neighbours in the whole graph.Therefore, S is both defensive r-alliance and offensive r-alliance inG .
2. If S ⊂ V is a global defensive r-alliance, then degS(v) ≥ deg S(v)+r ≥r, ∀v ∈ S. Moreover, if S ⊂ V is a global offensive (r + 2)-alliance,then degS(v) ≥ deg S(v) + r+ 2 ≥ r, ∀v ∈ S. Therefore, degS(v) ≥ r,∀v ∈ V .
Corollary 22. Each total k-dominating set is a global powerful r-alliance,where −∆ < r ≤ 2(k − 1)−∆.
Corollary 23.
• For −∆ < r ≤ 2k −∆, γkt
(G) ≥ γdr (G) and γkt
(G) ≥ γor (G).
• For −∆ < r ≤ 2(k − 1)−∆, γkt
(G) ≥ γ∗r (G).
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• For k ≥ 1, γ∗k(G) ≥ γkt
(G).
By Corollary 23, we have that lower bounds for γdr (G), γor (G) andγ∗r (G) lead to lower bounds for γ
kt(G). Moreover, upper bounds for γ
kt(G)
lead to upper bounds for γdr (G), γor (G) and γ∗r (G).
Theorem 24. [10] Let ϕ be the Laplacian spectral radius of G. Then
γor (G) ≥⌈nϕ
⌈δ+r
2
⌉⌉.
Corollary 25. Let ϕ be the Laplacian spectral radius of G. If r+∆2 ≤ k,
then γkt
(G) ≥⌈nϕ
⌈δ+r
2
⌉⌉.
5 Conclusions
We introduced and discussed the concept of global powerful r-alliances fromthe viewpoint of complexity as well as from the viewpoint of combinatorics.The latter study is well-motivated by the derived NP-hardness results. Ourstudy opens up quite natural lines of research, some of which we mentionin the following:
1. Does the mentioned NP-hardness result transfer to other, more re-stricted graph classes, e.g., to planar graphs? (The cases when r ≤ −2surely do so in Theorem 1, due to what is known about (total) dom-ination on planar cubic graph, see [21].) A similar question can beasked regarding k-total domination.
2. Conversely, are there interesting graph classes for which the men-tioned decision problems can be solved in polynomial time?
3. Similar questions can be stated in terms of approximability. (Noticethat the parameterized tractability results from [9] transfer to thealliance problems as discussed in this paper; however, approximabilityissues were not discussed as of yet.)
Also, many combinatorial questions remain. For instance, quite anumber of results are known for the graph parameter γt, while far lessis known for γkt for k > 1. To give some concrete research direction:Gravier [13] considered total domination on grid graphs. So, can we com-pute γkt(Pm�Pn) More generally speaking, upper bounds for γt(G) shouldprovide a generalized bound for γkt(G).
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Acknowledgments
We grateful acknowledge the comments of the referees that help improve thepresentation of the paper. This work was partly supported by the SpanishMinistry of Science and Innovation through projects MTM2009-07800, andby CONACYT, Mexico, through project CONACYT-UAG I0110/62/10.
References
[1] R. C. Brigham, R. D. Dutton and S. T. Hedetniemi. A sharp lowerbound on the powerful alliance number of Cm × Cn. Congressus Nu-merantium 167 (2004), 57–63.
[2] R. C. Brigham, R. D. Dutton, T. W. Haynes, and S. T. Hedetniemi.Powerful alliances in graphs. Discrete Mathematics 309 (2009), 2140–2147.
[3] A. Cami, H. Balakrishnan, N. Deo, and R. D. Dutton. On the com-plexity of finding optimal global alliances. Journal of CombinatorialMathematics and Combinatorial Computing 58 (2006), 23–31.
[4] M. Chellali and T. Haynes. Global alliances and independence in trees.Discussiones Mathematicae Graph Theory 27 (1) (2007), 19–27.
[5] E. J. Cockayne, R. Dawes and S. T. Hedetniemi. Total domination ingraphs. Networks 10 (1980), 211–215.
[6] R. Downey and M. Fellows. Parameterized Complexity. Springer-Verlag, Berlin, Heidelberg, New York, 1999.
[7] O. Favaron, G. Fricke, W. Goddard, S. M. Hedetniemi, S. T. Hedet-niemi, P. Kristiansen, R. C. Laskar, R. D. Skaggs. Offensive alliancesin graphs. Discussiones Mathematicae Graph Theory 24 (2) (2004),263–275.
[8] H. Fernau. Roman domination: a parameterized perspective. Inter-national Journal of Computer Mathematics 85 (2008), 25–38.
[9] H. Fernau and D. Raible. Alliances in graphs: a complexity-theoreticstudy. In J. van Leeuwen, G. F. Italiano, W. van der Hoek, C. Meinel,H. Sack, F. Plasil, and M. Bielikova, editors, SOFSEM 2007, Proceed-ings Vol. II, (2007), 61–70.
[10] H. Fernau, J. A. Rodrıguez and J.M. Sigarreta. Offensive k-alliancesin graphs. Discrete Applied Mathematics 157 (1) (2009), 177-182.
19
[11] John Frederick Fink and Michael S. Jacobson, n-domination in graphs,Graph theory with applications to algorithms and computer science(Kalamazoo, Mich., 1984), Wiley-Intersci. Publ., Wiley, New York,1985, pp. 283–300.
[12] M. R. Garey and D. S. Johnson. Computers and Intractability: A guideto the theory of NP-completeness, W. H. Freeman and co., New York,1979.
[13] S. Gravier. Total domination number of grid graphs. Discrete AppliedMathematics 121 (2002), 119–128.
[14] F. Harary and T. W. Haynes. Double domination in graphs. Ars Com-binatoria 55 (2000), 201–213.
[15] T. W. Haynes, S. T. Hedetniemi and M. A. Henning. Global defensivealliances in graphs. Electronic Journal of Combinatorics 10 (2003),139–146.
[16] L. H. Jamieson, S. T. Hedetniemi, and A. A. McRae. The algorithmiccomplexity of alliances in graphs. Journal of Combinatorial Mathe-matics and Combinatorial Computing 68 (2009), 137–150.
[17] L. H. Jamieson. Algorithms and Complexity for Alliances and WeightedAlliances of Different Types. PhD thesis, Clemson University, 2007.
[18] M. A. Henning and A. P. Kazemi, k-tuple total domination in graphs.Discrete Applied Mathematics 158 (2010) 1006–1011.
[19] T. Kikuno, N. Yoshida, and Y. Kakuda. The NP-Completeness of thedominating set problem in cubic planer graphs. IEICE TransactionsE63-E (1980), 443–444.
[20] P. Kristiansen, S. M. Hedetniemi and S. T. Hedetniemi. Alliancesin graphs. Journal of Combinatorial Mathematics and CombinatorialComputing 48 (2004), 157–177.
[21] D. F. Manlove. Minimaximal and maximinimal optimisation problems:a partial order-based approach. PhD thesis, University of Glasgow,Computing Science, 1998.
[22] R. Raz and S. Safra. A sub-constant error-probability low-degree test,and a sub-constant error-probability PCP characterization of NP. InProceedings of the 29th ACM Symposium on Theory of Computing,pages 475–484. ACM Press, 1997.
[23] J. A. Rodrıguez and J. M. Sigarreta. Offensive alliances in cubicgraphs. International Mathematical Forum 1 (36) (2006), 1773–1782.
20
[24] J. A. Rodrıguez and J. M. Sigarreta. Spectral study of alliances ingraphs. Discussiones Mathematicae Graph Theory 27 (1) (2007), 143–157.
[25] J. A. Rodrıguez-Velazquez, J. M. Sigarreta. Global alliances in planargraphs. AKCE International Journal on Graphs and Combinatorics 4(1) (2007), 83–98.
[26] J. A. Rodrıguez-Velazquez and J. M. Sigarreta. Global defensive k-alliances in graphs. Discrete Applied Mathematics 157 (2) (2009), 211–218.
[27] J. A. Rodrıguez-Velazquez, I. G. Yero and J. M. Sigarreta. Defensivek-alliances in graphs. Applied Mathematics Letters 22 (2009), 96–100.
[28] K. H. Shafique and R. D. Dutton. Maximum alliance-free and min-imum alliance-cover sets. Congressus Numerantium162 (2003), 139–146.
[29] K. H. Shafique and R. Dutton. A tight bound on the cardinalitiesof maximun alliance-free and minimun alliance-cover sets. Journal ofCombinatorial Mathematics and Combinatorial Computing 56 (2006),139–145.
[30] K. H. Shafique, Partitioning a Graph in Alliances and its Applicationto Data Clustering. Ph. D. Thesis, University of Central Florida, 2004.
[31] K. H. Shafique and R. D. Dutton. On satisfactory partitioning ofgraphs. Congressus Numerantium 154 (2002), 183–194.
[32] J. M. Sigarreta, Alliances in Graphs. Ph.D. Thesis, Universidad CarlosIII de Madrid, 2007.
[33] J. M. Sigarreta and J. A. Rodrıguez. On defensive alliance and linegraphs. Applied Mathematics Letters 19 (12) (2006), 1345–1350.
[34] J. M. Sigarreta and J. A. Rodrıguez. On the global offensive alliancenumber of a graph. Discrete Applied Mathematics 157 (2) (2009), 219-226.
[35] I. G. Yero and J. A. Rodrıguez-Velazquez. Boundary defensive k-alliances in graphs. Discrete Applied Mathematics 158 (2010), 1205–1211.
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