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

Göring, Frank

Harant, Jochen

English

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On domination in graphs

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   TU Ilmenau | Universitätsbibliothek | ilmedia, 2020 http://www.tu-ilmenau.de/ilmedia Göring, Frank; Harant, Jochen: On domination in graphsURN: urn:nbn:de:gbv:ilm1-2020200110  Original published in: Discussiones mathematicae. Graph theory / Uniwersytet Zielonogórski, Wydział Matematyki, Informatyki i Ekonometrii. - Warsaw : De Gruyter Open. - 25 (2005), 1/2, p. 7-12. Original published: 2005 ISSN: 2083-5892 DOI: 10.7151/dmgt.1254 [Visited: 2020-01-17]     This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 Unported license.  To view a copy of this license, visit  http://creativecommons.org/licenses/BY-NC-ND/3.0/ Discussiones Mathematicae 7Graph Theory 25 (2005 ) 7–12ON DOMINATION IN GRAPHSFrank Go¨ringDepartment of MathematicsChemnitz University of TechnologyD–09107 Chemnitz, Germanye-mail: frank.goering@mathematik.tu-chemnitz.deandJochen HarantDepartment of MathematicsTechnical University of IlmenauD–98684 Ilmenau, Germanye-mail: harant@mathematik.tu-ilmenau.deAbstractFor a finite undirected graph G on n vertices two continuous op-timization problems taken over the n-dimensional cube are presentedand it is proved that their optimum values equal the domination num-ber γ of G. An efficient approximation method is developed and knownupper bounds on γ are slightly improved.Keywords: graph, domination.2000 Mathematics Subject Classification: 05C69.1. Introduction and ResultsFor 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, N2(i) = {k ∈ V | k ∈ ⋃j∈N(i)N(j) \({i} ∪N(i))}, di = |N(i)|, ti = |N2(i)|, δ = mini∈V di, and ∆ = maxi∈V di.A set D ⊆ V (G) is a dominating set of G if ({i}∪N(i))∩D 6= ∅ for everyi ∈ V . The minimum cardinality of a dominating set of G is the domination8 F. Go¨ring and J. Harantnumber γ of G. In [7] γ = minx1,...,xn∈[0,1]∑i∈V (xi+(1−xi)∏j∈N(i)(1−xj))was proved. With x1 = . . . = xn = 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)1δ+ 1(δ+1)δ+1δ)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 suchthat each component of G has domination number greater than 1. Withoutmentioning in each case, we will use di, ti ≥ 1 for i = 1, . . . , n if G ∈ Γ. Forx1, . . . , xn ∈ [0, 1] letfG(x1, . . . , xn) =∑i∈V(xi(1−( ∏j∈N(i)xj)(1−∏k∈N2(i)xk))+ (1− xi)∏j∈N(i)(1− xj))gG(x1, . . . , xn) = fG(x1, . . . , xn)−∑i∈V(11 + di(1− xi)( ∏j∈N(i)(1− xj))( ∏k∈N2(i)(1− xk))).Theorem 1. If G ∈ Γ thenγ = minx1,...,xn∈[0,1]fG(x1, . . . , xn) = minx1,...,xn∈[0,1]gG(x1, . . . , xn)≤ minx∈[0,1]∑i∈V(x(1− xdi(1− xti))+ (1− x)di+1(1− 11 + di(1− x)ti))≤ minx∈[0,1](x(1− x∆(1− x))+ (1− x)δ+1(1− 11 + ∆(1− x)∆(∆−1)))n.Since DOMINATING SET is an NP-complete decision problem ([5]), it isdifficult to solve the continuous optimization problem P :minx1,...,xn∈[0,1]gG(x1, . . . , xn).However, if (x1, . . . , xn) is the solution of any approximation method for P,then (see Theorem 2) we can easily find a dominating set of G of cardinalityat most gG(x1, . . . , xn).On Domination in Graphs 9Theorem 2. Given a graph G ∈ Γ on V = {1, . . . , n} with maximum degree∆, x1, . . . , xn ∈ [0, 1], there is an O(∆4n)-algorithm finding a dominatingset D of G with |D| ≤ gG(x1, . . . , xn).2. ProofsProof of Theorem 1. For events A and B and for a random variable Zof 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 expectationof Z, respectively. Let A be the complementary event of A. We will usethe 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′|) =∑s∈S P (s ∈ S′) for a random subset S′ ofa given finite set S. I ⊂ V is an independent set if N(i)∩ I = ∅ for all i ∈ I.Consider fixed x1, . . . , xn ∈ [0, 1]. X ⊆ V is formed by random and indepen-dent choice of i ∈ V , where P (i ∈ X) = xi. Let X ′ = {i ∈ X | N(i) ⊆ X},X ′′ = {i ∈ X ′ | N(i) ∩ (X \X ′) 6= ∅}, Y = {i ∈ V | i /∈ X ∧N(i) ∩X = ∅},Y ′ = {i ∈ Y | N(i) ∩ Y 6= ∅}, and I be a maximum independent set of thesubgraph of G induced by Y ′.Lemma 3. (X \X ′′) ∪ (Y \ I) is a dominating set of G.Proof. Obviously, X ′′ ⊆ X ′ ⊆ X and (X\X ′) ⊆ (X\X ′′). If i ∈ V \(X∪Y )then N(i) ∩ (X \X ′) 6= ∅, if i ∈ X ′′ then again N(i) ∩ (X \X ′) 6= ∅, and ifi ∈ I then N(i) ∩ (Y \ I) 6= ∅.Lemma 4. γ ≤ E(|X|)− E(|X ′′|) + E(|Y |)− E(|I|).Proof. Let D be a random dominating set of G. Because of the property ofthe expectation to be an average value we have γ ≤ E(|D|). With Lemma 3and linearity of the expectation, γ ≤ E(|(X\X ′′)∪(Y \I)|) = E(|X|−|X ′′|+|Y |− |I|) = E(|X|)−E(|X ′′|)+E(|Y |)−E(|I|) since (X \X ′′)∩ (Y \ I) = ∅.Lemma 5. E(|X|) =∑i∈Vxi, E(|X ′′|) =∑i∈Vxi( ∏j∈N(i)xj)(1−∏k∈N2(i)xk),E(|Y |) =∑i∈V(1− xi)∏j∈N(i)(1− xj), andE(|I|) ≥∑i∈V11 + di(1− xi)( ∏j∈N(i)(1− xj))( ∏k∈N2(i)(1− xk)).10 F. Go¨ring and J. HarantProof. E(|X|) =∑i∈V P (i ∈ X) =∑i∈V xi.E(|X ′′|) =∑i∈VP (i ∈ X ′′) =∑i∈VP (i ∈ X ∧N(i) ⊆ X ∧N(i) ∩ (X \X ′) 6= ∅)=∑i∈VP (i ∈ X)P (N(i) ⊆ X)P (N(i) ∩ (X \X ′) 6= ∅ | i ∈ X ∧N(i) ⊆ X)=∑i∈Vxi( ∏j∈N(i)xj)(1− P (N(i) ⊆ X ′ | i ∈ X ∧N(i) ⊆ X))=∑i∈Vxi( ∏j∈N(i)xj)(1− P (N2(i) ⊆ X)) =∑i∈Vxi( ∏j∈N(i)xj)(1−∏k∈N2(i)xk).E(|Y |) =∑i∈VP (i ∈ Y ) =∑i∈VP (i /∈ X)P (N(i) ∩X = ∅)=∑i∈V(1− xi)∏j∈N(i)(1− xj).A lower bound on |I| (see [1, 9, 2, 6]) is given by the following inequality|I| ≥∑i∈Y ′ 11+di . For i ∈ V (G) define the random variable Zi with Zi = 11+diif i ∈ Y ′ and Zi = 0 if i /∈ Y ′. Hence,E(|I|) ≥ E(∑i∈VZi)=∑i∈VE(Zi) =∑i∈V11 + diP (i ∈ Y ′)=∑i∈V11 + diP (i /∈ X ∧N(i) ∩X = ∅ ∧N(i) ∩ Y 6= ∅).Because di ≥ 1, N2(i) ∩X = ∅ implies N(i) ∩ Y 6= ∅. Hence,E(|I|) ≥∑i∈V11 + diP (i /∈ X ∧N(i) ∩X = ∅ ∧N2(i) ∩X = ∅)=∑i∈V11 + diP (i /∈ X)P (N(i) ∩X = ∅)P (N2(i) ∩X = ∅)=∑i∈V11 + di(1− xi)( ∏j∈N(i)(1− xj))( ∏k∈N2(i)(1− xk)).On Domination in Graphs 11From Lemma 4 and Lemma 5 we have γ ≤ gG(x1, . . . , xn) ≤ fG(x1, . . . , xn).Let D∗ be a minimum dominating set of G and let yi = 1 if i ∈ D∗ andyi = 0 if i /∈ D∗. Then yi∏j∈N(i) yj = 0 and (1− yi)∏j∈N(i)(1− yj) = 0 forevery i ∈ V , γ = |D∗| =∑i∈V yi = gG(y1, . . . , yn) = fG(y1, . . . , yn), and theproof of Theorem 1 is complete.Proof of Theorem 2. Given a graph H on nH vertices with mH edges,there is an O(nH +mH)-algorithm A finding an independent set of H withcardinality at least∑y∈V (H)11+dH(y), where dH(y) is the degree of y ∈ V (H)in H (see [2]).First we present an algorithm that constructs a set D ⊆ V .AlgorithmINPUT: a graph G ∈ Γ on V = {1, . . . , n}, x1, . . . , xn ∈ [0, 1]OUTPUT: D(1) For l = 1, . . . , n do if ∂gG(x1,...,xn)∂xl ≥ 0 then xl := 0 else xl := 1.(2) X := {l ∈ {1, . . . , n} | xl = 1}. Calculate X ′′,Y ,Y ′, and I using A.(3) D := (X \X ′′) ∪ (Y \ I).ENDLet g∗ = gG(x1, . . . , xn), where (x1, . . . , xn) is the input vector. Note thatthe function gG is linear in each variable. Thus, in step (1), for fixedx1, . . . , xl−1, xl+1, . . . , xn we always choose xl in such a way that the valueof gG(x1, . . . , xn) is not increased. Hence, xl ∈ {0, 1} for l = 1, . . . , n andgG(x1, . . . , xn) ≤ g∗ after step (1) of the algorithm. With Lemma 3, D is adominating set, and with |S| = E(|S|) for a deterministic set S and Lemma5, |D| ≤ g∗. It is easy to see that ∂gG(x1,...,xn)∂xl can be calculated in O(∆4)time. Since G has O(∆n) edges, the algorithm runs in O(∆4n) time.References[1] Y. Caro, New results on the independence number (Technical Report, Tel-AvivUniversity, 1979).[2] Y. Caro and Zs. Tuza, Improved lower bounds on k-independence, J. GraphTheory 15 (1991) 99–107.[3] R. Diestel, Graph Theory, Graduate Texts in Mathematics (Springer, 1997).[4] N. Alon, J.H. Spencer and P. Erdo¨s, The Probabilistic Method (John Wileyand Sons, Inc. 1992), page 6.12 F. Go¨ring and J. Harant[5] M.R. Garey and D.S. Johnson, Computers and Intractability, A Guide to theTheory 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 independentsets of graphs, Combinatorics, Probability and Computing 8 (1999) 547–553.[8] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Dominationin 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 (BellLaboratories Technical Memorandum 81-11217-9, Murray Hill, NJ, 1981).Received 23 September 2003Revised 15 June 2004Powered by TCPDF (www.tcpdf.org)

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On domination in graphs

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On domination in graphs

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