ON DOMINATION IN GRAPHS

ON DOMINATION IN GRAPHS

Frank G¨ oring Department of Mathematics Chemnitz University of Technology

D–09107 Chemnitz, Germany

e-mail: frank.goering@mathematik.tu-chemnitz.de and

Jochen Harant Department of Mathematics Technical University of Ilmenau

D–98684 Ilmenau, Germany e-mail: harant@mathematik.tu-ilmenau.de

Abstract

For a finite undirected graph G on n vertices two continuous op- timization problems taken over the n-dimensional cube are presented and it is proved that their optimum values equal the domination num- ber γ of G. An efficient approximation method is developed and known upper bounds on γ are slightly improved.

Keywords: graph, domination.

2000 Mathematics Subject Classification: 05C69.

1. Introduction and Results

For terminology and notation not defined here we refer to [3]. Let V = V (G) = {1, . . . , n} be the vertex set of an undirected graph G, and for i ∈ V , N (i) be the neighbourhood of i in G, N ₂ (i) = {k ∈ V | k ∈ ^S _{j∈N (i)} N (j) \ ({i} ∪ N (i))}, d _i = |N (i)|, t _i = |N ₂ (i)|, δ = min _i∈V d _i , and ∆ = max _i∈V d _i .

A set D ⊆ V (G) is a dominating set of G if ({i}∪N (i))∩D 6= ∅ for every

i ∈ V . The minimum cardinality of a dominating set of G is the domination

number γ of G. In [7] γ = min _x

_,...,x

_∈[0,1] ^P _i∈V (x _i +(1−x _i ) ^Q _{j∈N (i)} (1−x _j )) was proved. With x ₁ = . . . = x _n = x we have γ ≤ (x + (1 − x) ^δ+1 )n ≤ (x+e ^−(δ+1)x )n for every x ∈ [0, 1]. Minimizing x+(1−x) ^δ+1 and x+e ^−(δ+1)x , the well-known inequalities γ ≤ (1 − ¹

(δ+1)

+ ¹

(δ+1)

)n ≤ ^1+ln(δ+1) _δ+1 n (see [4, 8]) follow. Obviously, it is easily checked whether γ = 1 or not. Thus, we will assume G ∈ Γ in the sequel, where Γ is the set of graphs G such that each component of G has domination number greater than 1. Without mentioning in each case, we will use d _i , t _i ≥ 1 for i = 1, . . . , n if G ∈ Γ. For x ₁ , . . . , x _n ∈ [0, 1] let

f _G (x ₁ , . . . , x _n ) = ^X

i∈V

Ã

x _i ^³ 1− ^{³ Y}

j∈N (i)

x _j ^´³ 1 − ^Y

k∈N

(i)

x _k ^´´ + (1 − x _i ) ^Y

j∈N (i)

(1 − x _j )

!

g _G (x ₁ , . . . , x _n ) = f _G (x ₁ , . . . , x _n )

− ^X

i∈V

à 1

1 + d _i (1 − x _i ) ^{³ Y}

j∈N (i)

(1 − x _j ) ^{´³ Y}

k∈N

(i)

(1 − x _k ) ^´

! .

Theorem 1. If G ∈ Γ then

γ = min

x

,...,x

∈[0,1] f _G (x ₁ , . . . , x _n ) = min

x

,...,x

∈[0,1] g _G (x ₁ , . . . , x _n )

≤ min

x∈[0,1]

X

i∈V

Ã

x ^³ 1 − x ^d

(1 − x ^t

) ^´ + (1 − x) ^d

^{+1 ³} 1 − 1

1 + d _i (1 − x) ^t

^´

!

≤ min

x∈[0,1]

Ã

x ^³ 1 − x ^∆ (1 − x) ^´ + (1 − x) ^{δ+1 ³} 1 − 1

1 + ∆ (1 − x) ^{∆(∆−1) ´}

! n.

Since DOMINATING SET is an NP-complete decision problem ([5]), it is difficult to solve the continuous optimization problem P :

x

,...,x min

∈[0,1] g _G (x ₁ , . . . , x _n ).

However, if (x ₁ , . . . , x _n ) is the solution of any approximation method for P,

then (see Theorem 2) we can easily find a dominating set of G of cardinality

at most g _G (x ₁ , . . . , x _n ).

Theorem 2. Given a graph G ∈ Γ on V = {1, . . . , n} with maximum degree

∆, x ₁ , . . . , x _n ∈ [0, 1], there is an O(∆ ⁴ n)-algorithm finding a dominating set D of G with |D| ≤ g _G (x ₁ , . . . , x _n ).

2. Proofs

P roof of T heorem 1. For events A and B and for a random variable Z of an arbitrary random space, P (A), P (A|B), and E(Z) denote the prob- ability of A, the conditional probability of A given B, and the expectation of Z, respectively. Let A be the complementary event of A. We will use the well-known facts that P (B)P (A|B) = P (A ∩ B) = P (B) − P (A ∩ B) = P (B)(1 − P (A|B)) and E(|S ⁰ |) = ^P _s∈S P (s ∈ S ⁰ ) for a random subset S ⁰ of a given finite set S. I ⊂ V is an independent set if N (i) ∩ I = ∅ for all i ∈ I.

Consider fixed x ₁ , . . . , x _n ∈ [0, 1]. X ⊆ V is formed by random and indepen- dent choice of i ∈ V , where P (i ∈ X) = x _i . Let X ⁰ = {i ∈ X | N (i) ⊆ X}, X ⁰⁰ = {i ∈ X ⁰ | N (i) ∩ (X \ X ⁰ ) 6= ∅}, Y = {i ∈ V | i / ∈ X ∧ N (i) ∩ X = ∅}, Y ⁰ = {i ∈ Y | N (i) ∩ Y 6= ∅}, and I be a maximum independent set of the subgraph of G induced by Y ⁰ .

Lemma 3. (X \ X ⁰⁰ ) ∪ (Y \ I) is a dominating set of G.

P roof. Obviously, X ⁰⁰ ⊆ X ⁰ ⊆ X and (X \X ⁰ ) ⊆ (X \X ⁰⁰ ). If i ∈ V \(X ∪Y ) then N (i) ∩ (X \ X ⁰ ) 6= ∅, if i ∈ X ⁰⁰ then again N (i) ∩ (X \ X ⁰ ) 6= ∅, and if i ∈ I then N (i) ∩ (Y \ I) 6= ∅.

Lemma 4. γ ≤ E(|X|) − E(|X ⁰⁰ |) + E(|Y |) − E(|I|).

P roof. Let D be a random dominating set of G. Because of the property of the expectation to be an average value we have γ ≤ E(|D|). With Lemma 3 and linearity of the expectation, γ ≤ E(|(X \X ⁰⁰ )∪(Y \I)|) = E(|X|−|X ⁰⁰ |+

|Y | − |I|) = E(|X|) − E(|X ⁰⁰ |) + E(|Y |) − E(|I|) since (X \ X ⁰⁰ ) ∩ (Y \ I) = ∅.

Lemma 5. E(|X|) = ^X

i∈V

x _i , E(|X ⁰⁰ |) = ^X

i∈V

x _i ^{³ Y}

j∈N (i)

x _j ^´³ 1 − ^Y

k∈N

(i)

x _k ^´ ,

E(|Y |) = ^X

i∈V

(1 − x _i ) ^Y

j∈N (i)

(1 − x _j ), and

E(|I|) ≥ ^X

i∈V

1

1 + d _i (1 − x _i ) ^{³ Y}

j∈N (i)

(1 − x _j ) ^{´³ Y}

k∈N

(i)

(1 − x _k ) ^´ .

P roof. E(|X|) = ^P _i∈V P (i ∈ X) = ^P _i∈V x _i . E(|X ⁰⁰ |) = ^X

i∈V

P (i ∈ X ⁰⁰ ) = ^X

i∈V

P (i ∈ X ∧ N (i) ⊆ X ∧ N (i) ∩ (X \ X ⁰ ) 6= ∅)

= ^X

i∈V

P (i ∈ X)P (N (i) ⊆ X)P (N (i) ∩ (X \ X ⁰ ) 6= ∅ | i ∈ X ∧ N (i) ⊆ X)

= ^X

i∈V

x _i ^{³ Y}

j∈N (i)

x _j ^´ (1 − P (N (i) ⊆ X ⁰ | i ∈ X ∧ N (i) ⊆ X))

= ^X

i∈V

x _i ^{³ Y}

j∈N (i)

x _j ^´ (1 − P (N ₂ (i) ⊆ X)) = ^X

i∈V

x _i ^{³ Y}

j∈N (i)

x _j ^´³ 1 − ^Y

k∈N

(i)

x _k ^´ .

E(|Y |) = ^X

i∈V

P (i ∈ Y ) = ^X

i∈V

P (i / ∈ X)P (N (i) ∩ X = ∅)

= ^X

i∈V

(1 − x _i ) ^Y

j∈N (i)

(1 − x _j ).

A lower bound on |I| (see [1, 9, 2, 6]) is given by the following inequality

|I| ≥ ^P _i∈Y

1

1+d

. For i ∈ V (G) define the random variable Z _i with Z _i = _1+d ¹

if i ∈ Y ⁰ and Z _i = 0 if i / ∈ Y ⁰ . Hence,

E(|I|) ≥ E ^{³ X}

i∈V

Z _i ^´ = ^X

i∈V

E(Z _i ) = ^X

i∈V

1

1 + d _i P (i ∈ Y ⁰ )

= ^X

i∈V

1

1 + d _i P (i / ∈ X ∧ N (i) ∩ X = ∅ ∧ N (i) ∩ Y 6= ∅).

Because d _i ≥ 1, N ₂ (i) ∩ X = ∅ implies N (i) ∩ Y 6= ∅. Hence, E(|I|) ≥ ^X

i∈V

1

1 + d _i P (i / ∈ X ∧ N (i) ∩ X = ∅ ∧ N ₂ (i) ∩ X = ∅)

= ^X

i∈V

1

1 + d _i P (i / ∈ X)P (N (i) ∩ X = ∅)P (N ₂ (i) ∩ X = ∅)

= ^X

i∈V

1

1 + d _i (1 − x _i ) ^{³ Y}

j∈N (i)

(1 − x _j ) ^{´³ Y}

k∈N

(i)

(1 − x _k ) ^´ .

From Lemma 4 and Lemma 5 we have γ ≤ g _G (x ₁ , . . . , x _n ) ≤ f _G (x ₁ , . . . , x _n ).

Let D ^∗ be a minimum dominating set of G and let y _i = 1 if i ∈ D ^∗ and y _i = 0 if i / ∈ D ^∗ . Then y _i ^Q _{j∈N (i)} y _j = 0 and (1 − y _i ) ^Q _{j∈N (i)} (1 − y _j ) = 0 for every i ∈ V , γ = |D ^∗ | = ^P _i∈V y _i = g _G (y ₁ , . . . , y _n ) = f _G (y ₁ , . . . , y _n ), and the proof of Theorem 1 is complete.

P roof of T heorem 2. Given a graph H on n _H vertices with m _H edges, there is an O(n _H + m _H )-algorithm A finding an independent set of H with cardinality at least ^P _{y∈V (H) 1+d} ¹

H

(y) , where d _H (y) is the degree of y ∈ V (H) in H (see [2]).

First we present an algorithm that constructs a set D ⊆ V . Algorithm

INPUT: a graph G ∈ Γ on V = {1, . . . , n}, x ₁ , . . . , x _n ∈ [0, 1]

OUTPUT: D

(1) For l = 1, . . . , n do if ^∂g

^(x _∂x

^,...,x

⁾

l

≥ 0 then x _l := 0 else x _l := 1.

(2) X := {l ∈ {1, . . . , n} | x _l = 1}. Calculate X ⁰⁰ ,Y ,Y ⁰ , and I using A.

(3) D := (X \ X ⁰⁰ ) ∪ (Y \ I).

END

Let g ^∗ = g _G (x ₁ , . . . , x _n ), where (x ₁ , . . . , x _n ) is the input vector. Note that the function g _G is linear in each variable. Thus, in step (1), for fixed x ₁ , . . . , x _l−1 , x _l+1 , . . . , x _n we always choose x _l in such a way that the value of g _G (x ₁ , . . . , x _n ) is not increased. Hence, x _l ∈ {0, 1} for l = 1, . . . , n and g _G (x ₁ , . . . , x _n ) ≤ g ^∗ after step (1) of the algorithm. With Lemma 3, D is a dominating set, and with |S| = E(|S|) for a deterministic set S and Lemma 5, |D| ≤ g ^∗ . It is easy to see that ^∂g

^(x _∂x

^,...,x

⁾

l

can be calculated in O(∆ ⁴ ) time. Since G has O(∆n) edges, the algorithm runs in O(∆ ⁴ n) time.

References

[1] Y. Caro, New results on the independence number (Technical Report, Tel-Aviv University, 1979).

[2] Y. Caro and Zs. Tuza, Improved lower bounds on k-independence, J. Graph Theory 15 (1991) 99–107.

[3] R. Diestel, Graph Theory, Graduate Texts in Mathematics (Springer, 1997).

[4] N. Alon, J.H. Spencer and P. Erd¨os, The Probabilistic Method (John Wiley

and Sons, Inc. 1992), page 6.

[5] M.R. Garey and D.S. Johnson, Computers and Intractability, A Guide to the Theory of NP-Completeness (W.H. Freeman and Company, San Francisco, 1979).

[6] J. Harant, Some news about the independence number of a graph, Discuss.

Math. Graph Theory 20 (2000) 71–79.

[7] J. Harant, A. Pruchnewski and M. Voigt, On dominating sets and independent sets of graphs, Combinatorics, Probability and Computing 8 (1999) 547–553.

[8] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc., New York, N.Y., 1998), page 52.

[9] V.K. Wei, A lower bound on the stability number of a simple graph (Bell Laboratories Technical Memorandum 81-11217-9, Murray Hill, NJ, 1981).

Received 23 September 2003

Revised 15 June 2004