Upper bounds on the balanced 〈r,s〉 -domination number of a graph

Discrete Applied Mathematics 179 (2014) 214–221

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Upper bounds on the balanced ⟨r , s⟩-domination number of a graph A. Roux a,∗ , J.H. van Vuuren b a

Division of Applied Mathematics, Stellenbosch University, Private Bag X1, Matieland, 7602, South Africa

b

Department of Industrial Engineering, Stellenbosch University, Private Bag X1, Matieland, 7602, South Africa

article

info

Article history: Received 22 July 2013 Received in revised form 2 July 2014 Accepted 20 July 2014 Available online 7 August 2014 Keywords: s-dominating r-function ⟨r , s⟩-domination k-tuple domination Multiple domination Probabilistic method

abstract Let G = (V , E ) be a simple graph of order n with vertex set V = {v1 , . . . , vn } and suppose that at most ri units of some commodity may be placed at any vertex vi while at least si units must be placed in the closed neighbourhood of vi for i = 1, . . . , n. The smallest number of units that may be placed on the vertices of the graph satisfying the above requirements is called the ⟨r , s⟩-domination number of the graph. The case where r = [r , . . . , r ] and s = [s, . . . , s] is called the balanced case of ⟨r , s⟩-domination. We establish three upper bounds on the ⟨r , s⟩-domination number of a graph for the balanced case in this paper. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Let G = (V , E ) be a simple graph of order n with vertex set V = {v1 , . . . , vn } and define the open neighbourhood N (vi ) of a vertex vi ∈ V as the set {vj : vi vj ∈ E } and its closed neighbourhood N [vi ] as the set N (vi ) ∪ {vi }. Furthermore, let dv denote the degree of the vertex v . A set S ⊆ V is a dominating set of G if every vertex in V is either in S or adjacent to a vertex in S. The domination number γ (G) is the minimum cardinality of a dominating set of G. Cockayne [4] introduced a general framework for domination in graphs. Let r and s be non-negative integer n-vectors. As in [4], we define  an r-function of G as a function f : V → N0 satisfying f (vi ) ≤ ri for all i = 1, . . . , n. For every v ∈ V , let f [v] = f is called s-dominating if f [vi ] ≥ si for each i = 1, . . . , n. The weight u∈N [v] f (u). An r-function  of an r-function f is defined as |f | = f (v) . The smallest weight of an s-dominating r-function is called the ⟨r , s⟩v∈V s domination number of G and is denoted by γ ( G ). As noted in [4], an s-dominating r-function of G exists if and only if r  vj ∈N [vi ] rj ≥ si for all i = 1, . . . , n. Note that if r = s = [1, . . . , 1], then the ⟨r , s⟩-domination number of G is the (classical) domination number γ (G) of G. Other special cases of ⟨r , s⟩-domination include {k}-domination [2,11,20] and k-tuple domination [12,15,18,7] which are the cases where r = s = [k, . . . , k] and where r = [1, . . . , 1] and s = [k, . . . , k], respectively, for some k ∈ N. The k-tuple domination number of a graph is denoted by γ×k (G). The balanced case of ⟨r , s⟩-domination, where r = [r , . . . , r ] and s = [s, . . . , s], was studied in [21,19] for r , s ∈ N. It is well known that the problem of computing the (classical) domination number of a graph, a special case of ⟨r , s⟩domination, is NP-complete [8]. Another special case, the problem of computing the k-tuple domination number, is also

∗

Corresponding author. Tel.: +27 722000083; fax: +27 21 808 3406. E-mail addresses: [email protected], [email protected] (A. Roux), [email protected] (J.H. van Vuuren).

http://dx.doi.org/10.1016/j.dam.2014.07.016 0166-218X/© 2014 Elsevier B.V. All rights reserved.

A. Roux, J.H. van Vuuren / Discrete Applied Mathematics 179 (2014) 214–221

215

NP-complete, even for split graphs [12]. These complexity results suggest that the problem of computing the balanced case of ⟨r , s⟩-domination may be NP-complete for arbitrary values of r and s. Alon and Spencer [1] used the probabilistic method in 1991 to show that

γ (G) ≤

n(1 + ln(δ + 1))

(1)

δ+1

for a graph G of order n with minimum degree δ . This result was, in fact, independently established earlier, without the use of the probabilistic method, by Payan [17] and Lovasz [13] in 1975. Harant et al. [10] improved this bound and showed that

 γ (G) ≤ n 1 − δ



1+1/δ 

1

δ+1

.

(2)

Henning and Harant [9] employed the same method in 2005 to prove that

γ×2 (G) ≤ n



ln(1 + d) + ln δ + 1



δ

,

where d = 1n v∈V dv . This idea was further generalised in 2007 to k-tuple domination with the following conjecture by Rautenbach and Volkmann [18].



Conjecture 1. If k ∈ N and G is a graph of order n with minimum degree δ ≥ k, then

    dv + 1  − ln n + 1 . ln(δ + 2 − k) + ln γ×k (G) ≤ k−1 δ+2−k v∈V 

n



(3)

The special case of Conjecture 1 where k = 3 was established by Rautenbach and Volkmann [18] themselves, while the general conjecture was proven by Chang [3], Xu et al. [23] and Zverovich [24] independently in 2008. The results of Chang [3] and Zverovich [24] are based on the model and approach presented originally in [7], which was further developed in [5]. Rautenbach and Volkmann [18] also established another upper bound on k-tuple domination which does not depend on the degree sequence of G, namely

γ×k (G) ≤



n

δ+1

k ln(δ + 1) +

k−1  i =0

(k − i) i!(δ + 1)k−i−1

 ,

(4)

where G is a graph of order n with minimum degree δ with 2k ≤ (δ + 1)/ ln(δ + 1). Recently, Gagarin et al. [6] improved the bounds (3) and (4) and proposed a bound, generalised from (1) and (2), which also does not depend on the degree sequence of G. They showed that

 γ×k (G) ≤ n 1 −



δ′ 1/δ ′

b˜ k−1 (1 + δ ′ )1+1/δ

(5)

′

for any graph G with minimum degree δ ≥ k, where δ ′ = δ − k + 1 and b˜ t =



δ+1 t



.

Another probabilistic bound, again independent of the degree sequence of G, was proposed by Przybyło [16] in 2013, who showed that

 γ×k (G) ≤ n

k  ln(δ + 2 − i) + 1 i=1

δ+2−i

 (6)

for any graph G with minimum degree δ ≥ k − 1, where k ≤ (δ + 2 − k)/(ln(δ + 2 − k) + 1). After introducing three new bounds on the ⟨r , s⟩-domination number in Section 2 we present two ways to improve these bounds in Section 3. The paper closes in Section 4 with an experimental comparison of the new bounds with those in the literature, showing that none of the new bounds always outperforms the others. Furthermore, for some graphs, one of the bounds presented in Section 2 provides better results for k-tuple domination than the existing bound in (5) for certain values of k. 2. The bounds In this section, three bounds on the ⟨r , s⟩-domination number of a graph are established for the case where r = [r , . . . , r ] and s = [s, . . . , s]. Let G be a graph of order n with minimum degree δ and let r = [r , . . . , r ] and s = [s, . . . , s] for some r , s ∈ N such that n r (δ + 1) ≥ s. The function f (v) = r, v ∈ V (G) is clearly an s-dominating r-function of G and therefore i=1 ri = rn is an upper bound on the ⟨r , s⟩-domination number of G. However, this bound can always be improved since γrs (G) ≤ γrs′ (G) where r ′ = [r − 1, . . . , r − 1].

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Theorem 1. Let G be a graph of order n with minimum degree δ and let r = [r , . . . , r ] and s = [s, . . . , s] for some r , s ∈ N such that r (δ + 1) ≥ s. Then

γ (G) ≤ s r





s

δ+1

n.

Proof. For any graph triple (G, r , s), γrs (G) ≤ γrs′ (G) where r ′ = [r − 1, . . . , r − 1] [19]. Hence

γss (G) ≤ γss−1 (G) ≤ · · · ≤ γrsmin (G), where rmin = ⌈s/(δ + 1)⌉. Furthermore, since the function f (v) = rmin , v ∈ V (G) is an s-dominating function of G, it follows that

γ (G) ≤ γ s r

s rmin



(G) ≤



s

δ+1

n. 

The bound in Theorem 1 is sharp and is achieved only when G is a regular graph and r = s/(δ + 1). For the second bound an ⟨r , s⟩-dominating set is constructed by first choosing vertices randomly from a multiset containing r copies of each vertex v ∈ V and then adding the necessary number of vertices to ensure that the chosen set is indeed s-dominating. This set then provides an upper bound on the ⟨r , s⟩-domination number of the graph. The resulting bound is a generalisation of the 2013 bound on the k-tuple domination number in (5) by Gagarin et al. [6]. Let A, B and C be multisets over V where f (v) and g (v) denote the multiplicity of v in A and B, respectively. The sum of two multisets A and B, denoted by A ⊎ B, is defined as the multiset C with multiplicity f (v) + g (v) for all v ∈ V . The removal of the multiset B from A, denoted by A ⊖ B, is defined as the multiset C with multiplicity max{(f (v) − g (v)), 0} for all v ∈ V . Theorem 2. Let G be a graph of order n with minimum degree δ and let r = [r , . . . , r ] and s = [s, . . . , s] for some r , s ∈ N such that r (δ + 1) ≥ s. Then

 γ (G) ≤ rn 1 − d s r

where mi =



r (δ+1) i





1/d 

r

1+1/d 

1 d+1

ms−1

,

and d = r (δ + 1) − s.

Proof. For each v ∈ V , as in [6], let N ′ (v) denote a set containing δ vertices of N (v) and let N ′ [v] = N ′ (v) ∪ {v}. Let U be the multiset over V containing r copies of each vertex v ∈ V and form a multiset D by picking every element u ∈ U independently at  random with Pr(u ∈ D) = p for some 0 < p ≤ 1. Let f (u) denote the resulting multiplicity of u in D and let Di = {v ∈ V | u∈N ′ [v] f (u) = i} for i = 0, . . . , s − 1. Then, Pr(v ∈ Di ) =



r (δ + 1)



i

pi (1 − p)r (δ+1)−i .

Now, since r (δ + 1) ≥ s there exists a subset D′i ⊆ U ⊖ D such that |D′i | ≤ (s − i)|Di | and u∈N [v] g (u) ≥ s − i for every v ∈ Di , where g (u) denotes the resulting multiplicity of u in D′i . For example, let Dvi be a multiset containing s − i elements of U ⊖ D that is in the closed neighbourhood of v for every v ∈ Di . Furthermore, let D′i be the union of all the multisets Dvi such that no element appears more than r times. Then D′i has the desired properties. −1 ′ Let A = D ⊎ (⊎si= 0 Di ) and define the function h such that h(v) is the minimum of the multiplicity of v in A and r. Then h is an s-dominating r-function of G and it follows that



γrs (G) ≤ E (|h|) ≤ E (|A|) ≤ E (|D|) +

s−1 

E (|D′i |) ≤ E (|D|) +

i=0

s−1  (s − i)E (|Di |) i =0

by the linearity of expectation. Therefore,

γrs (G) ≤ E (|D|) +

s−1  (s − i)E (|Di |) i=0

≤



Pr(u ∈ D) +

u∈U

s−1   (s − i) Pr(v ∈ Di ) i=0

v∈V

s −1  ≤ rnp + n (s − i)mi pi (1 − p)r (δ+1)−i i =0

= rnp + n(1 − p)r (δ+1)−s+1

s−1  (s − i)mi pi (1 − p)s−1−i . i=0

A. Roux, J.H. van Vuuren / Discrete Applied Mathematics 179 (2014) 214–221

Now let h(p) = shown that

s−1 i =0

217

(s − i)mi pi (1 − p)s−1−i . To show that h(p) is monotonically non-decreasing in p for 0 < p ≤ 1 it is

h′ (p) = −s(s − 1)m0 (1 − p)s−2 + (s − 1)m1 (1 − p)s−2





  + −(s − 1)(s − 2)m1 p1 (1 − p)s−3 + 2(s − 2)m2 p1 (1 − p)s−3 .. .   + −2ms−2 ps−2 + (s − 1)ms−1 ps−2 s−2 

=

(−(s − i)mi + (i + 1)mi+1 ) (s − i − 1)pi (1 − p)s−i−2

i =0

is nonnegative. It suffices to show that −(s − i)mi + (i + 1)mi+1 ≥ 0 for 0 ≤ i ≤ s − 2. For these values of i,

    r (δ + 1) r (δ + 1) (i + 1)mi+1 − (s − i)mi = (i + 1) − (s − i) i+1 i      r (δ + 1) r (δ + 1) = (i + 1) − (s − i) i+1 i   r (δ + 1) = (r (δ + 1) − s) . i

Since r (δ + 1) ≥ s it is clear that −(s − i)mi + (i + 1)mi+1 ≥ 0. Hence

γrs (G) ≤ rnp + (1 − p)r (δ+1)−s+1 h(p) ≤ rnp + (1 − p)r (δ+1)−s+1 h(1) = rnp + (1 − p)d+1 ms−1 . The expression rnp + (1 − p)d+1 ms−1 is minimised for

 p=1−

1/d

r ms−1 (d + 1)

,

which shows that

 γ (G) ≤ rn 1 − d s r



1/d 

r

1+1/d 

1 d+1

ms−1

. 

For the next bound an r-function is constructed where each function value is chosen randomly from the set {0, . . . , r }; thereafter, the function values are increased, where necessary, to ensure that the function is s-dominating. Theorem 3. Let G be a graph of order n with minimum degree δ and let r = [r , . . . , r ] and s = [s, . . . , s] for some r , s ∈ N such that r (δ + 1) ≥ s. Then rn

γ (G) ≤ s r

2

+

s−1 

(s − i)

v +1  d

(−1)

v∈V k=0

i=0

k



dv + 1 k



i + dv − k(r + 1) dv



(r + 1)−dv −1 .

Proof. Let f be a function on V which assigns values in {0, . . . , r } to each vertex v ∈ V independently with probability p = r +1 1 . Let Di = {v ∈ V | f [u] = i} for i = 0, . . . , s − 1. Since r (δ + 1) ≥ s, there exists a function gi such that |gi | ≤ (s − i)|Di | and gi [v] ≥ s − i for each v ∈ Di . For example, let giv be a function such that giv [v] = s − i for every v ∈ Di and let gi be a function such that gi (v) = min{ v∈Di gi (v), r }. Then gi has the desired properties. Let a denote the number of non-negative integer solutions of the equation x1 + · · · + xdv +1 = i with xj ≤ r. Then a is the coefficient of z i [14] in the generating function



1 − z r +1

dv

1−z and hence dv +1

a=

 k=0

(−1)

k



dv + 1



i + dv − k(r + 1)

k

It follows that Pr(v ∈ Di ) = apdv +1 .

dv



.

218

A. Roux, J.H. van Vuuren / Discrete Applied Mathematics 179 (2014) 214–221

Fig. 1. A comparison of the bounds in Section 2 for the random graph G1 in [22] of order 50 with δ = 5, ∆ = 40 and average degree = 22.6. Illustrated above are the cases where r = 1, r = 10 and r = 20. The value of γrs (G1 ) lies with in the shaded region.

Let h be a function on V such that h(v) = min{f (v) + that

γrs (G) ≤ E (|h|) ≤ E (|f |) +

s−1 

E (|gi |) ≤



s−1 i=0

E (f (v)) +

s−1 

v∈V

i=0

gi (v), r }. Then h is an s-dominating r-function and it follows

(s − i)E (|Di |).

(7)

i=0

Furthermore, the expected value of f (v) is E (f (v)) =

r 

j Pr(f (v) = j) =

r 

j=0

jp =

r (r + 1)

j=0

2

p=

r 2

.

(8)

Moreover, the expected value of the size of Di is E (|Di |) =



Pr(v ∈ Di ) =

v∈V



apdv +1

v∈V

   v +1  d dv + 1 i + dv − k(r + 1) k = (−1) (r + 1)−dv −1 . v∈V k=0

k

dv

The desired bound follows from substituting (8)–(9) into (7).

(9)



In the classical domination setting (i.e. where r = s = 1) the bound in Theorem 3 reduces to a bound strictly larger than n/2 for a graph of order n. However, in the case of the k-tuple domination number (i.e. where r = 1 and s = k) the bound in Theorem 3 outperforms the bound (5) from [6] for some graphs and for large values of k. 3. General improvements All of the bounds in Section 2 may possibly be improved in two ways. Let the vectors r = [r1 , . . . , rn ] and s = [s1 , . . . , sn ] satisfy vj ∈N [vi ] rj ≥ si for every vi ∈ V . First consider the case where mini {ri } > maxi {si }.

A. Roux, J.H. van Vuuren / Discrete Applied Mathematics 179 (2014) 214–221

219

Fig. 2. A comparison of the bounds in Section 2 for the random graph G2 in [22] of order 100 with δ = 19, ∆ = 43 and average degree = 30. Illustrated above are the cases where r = 1, r = 5 and r = 10. The last plot shows the bounds without the implementation of Lemmas 4 and 5 for the case where r = 10. The value of γrs (G2 ) lies with in the shaded region.

Lemma 4. Let G be a graph of order n with minimum degree δ and let the vectors r = [r1 , . . . , rn ] and s = [s1 , . . . , sn ] satisfy  vj ∈N [vi ] rj ≥ si for every vi ∈ V . Furthermore, let rmin = mini {ri } and smax = maxi {si }. If rmin > smax , then

γrs (G) = γss′ (G), where s′ = {smax , . . . , smax }. Proof. Let f be an s-dominating s′ -function of G of minimum weight. Since rmin > smax , f is also an s-dominating r-function of G and it follows that

γrs (G) ≤ |f | = γss′ (G).

(10)

Now let f be an s-dominating r-function of G of minimum weight such that there exists a vertex v ∈ V for which f (v) > smax . The function f ′ (u) =



smax f (u)

if u = v if u ̸= v

is an s-dominating r-function of G with |f ′ | < |f |. This contradicts the minimality of f and therefore f (v) ≤ smax for all v ∈ V . It follows that f is an s-dominating s′ -function and so

γss′ (G) ≤ |f | = γrs (G). The desired result follows by a combination of (10) and (11).

(11) 

By using Lemma 4, the value of r may be replaced with that of s, whenever r > s. The use of this lemma will always improve the bounds in Theorems 2 and 3. For the next improvement, only the balanced case where r = [r , . . . , r ] and s = [s, . . . , s] with r ≤ s is considered.

220

A. Roux, J.H. van Vuuren / Discrete Applied Mathematics 179 (2014) 214–221

Fig. 3. A comparison of the bounds in Section 2 for the 10-regular graph G3 in [22] of order 50. Illustrated above are the cases where r = 1, r = 10 and r = 20. The last plot shows the bounds without the implementation of Lemmas 4 and 5 for the case where r = 20. The value of γrs (G3 ) lies with in the shaded region.

Lemma 5. Let G be a graph of order n with minimum degree δ and maximum degree ∆, and let the vectors r = [r , . . . , r ] and s = [s, . . . , s] satisfy (δ + 1)r ≥ s. For vectors r ′ = [r ′ , . . . , r ′ ] and s′ = [s′ , . . . , s′ ], where r ′ = r − ⌊s/(∆ + 1)⌋ and s′ = s − (δ + 1) ⌊s/(∆ + 1)⌋, it follows that

γ (G) ≤ n s r



s

∆+1



′

+ γrs′ (G).

Proof. Let g be an s′ -dominating r ′ -function of G of minimum weight and define the function f such that f (v) = ⌊s/(∆ + 1)⌋ + g (v). Since g (v) ≤ r ′ = r − ⌊s/(∆ + 1)⌋, it follows that f (v) ≤ r. Furthermore, g [v] ≥ s′ = s − (δ + 1) ⌊s/(∆ + 1)⌋ and therefore f [v] =



f (v) = (dv + 1) ⌊s/(∆ + 1)⌋ + g [v] ≥ s.

u∈N [v]

Hence f is an s-dominating r-function of G and the desired bound follows.



The application of Lemma 5 to the bounds in Theorems 2 and 3 produces better results for values of s that are far greater than the value of r. 4. Bound comparison In this section three graphs are used to illustrate the relative performances of the new bounds in Section 2. It is shown, in particular, that none of these bounds always outperforms the others for these specific graphs. Note that in the case where r = s = 1, the bound of Theorem 2 is the same as the bound in (5) from [6]. To illustrate the effectiveness of these bounds,

A. Roux, J.H. van Vuuren / Discrete Applied Mathematics 179 (2014) 214–221

221

the lower bound γrs (G) ≥ ∆sn [21] is also included in all the comparisons, where G is a graph of order n with maximum +1 degree ∆. First consider the graph G1 of order 50 in [22] with minimum degree δ = 5, maximum degree ∆ = 40 and an average degree of 22.6. The graph G1 was constructed from a randomly generated degree sequence with values between 5 and 45. Fig. 1 depicts the bounds of Section 2 for three values of r over all possible s values. In the case of k-tuple domination, where r = 1 and s = k, the bound in Theorem 3 is the superior of the three bounds for all values of s, except for the special case where s = 1. When r = 10 or r = 20, the bound in Theorem 2 performs best for s = 1 and s = 2, while the bound in Theorem 3 performs the best when s > r (δ + 1)/2. For all the other values, the bound in Theorem 1 is the superior bound. Secondly, consider the graph G2 in [22] of order 100 with minimum degree δ = 19, maximum degree ∆ = 43 and an average degree of 30. The graph G2 was randomly generated with an edge probability of 0.3. Fig. 2 depicts the bounds of Section 2 for three values of r over all possible s values. In the case of k-tuple domination, where r = 1 and s = k, the bound in Theorem 2 is the superior of the bounds for values of s ≤ 4 or s ≥ 13, while the bound in Theorem 3 performs best for 4 < s < 13. When r = 5 or r = 10, the bound in Theorem 1 outperforms the bounds in Section 2 in most cases. The bound in Theorem 3 performs better than the bound in Theorem 1 for a few values of s, while the bound in Theorem 2 performs best when s is very small or when ⌈s/(δ + 1)⌉ = r. Finally, consider the 10-regular graph G3 in [22] with minimum degree δ = 10. Fig. 3 depicts the bounds of Section 2 for three values of r over all possible values of s. In the case of k-tuple domination, where r = 1 and s = k, the bound in Theorem 2 performs best, except for the cases where s = 3, 4; in these cases the bound in Theorem 3 performs better. When r = 10 or r = 20, the use of Lemma 5 renders a considerable improvement on the bound of Theorem 2. The bound in Theorem 1 performs the best for most values of s, with the bound in Theorem 2 outperforming the other bounds when the difference between s and c (δ + 1) is small for c ∈ N. 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]

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