A NOTE ON DOMINATION PARAMETERS IN RANDOM GRAPHS

Transcription

1 Discussioes Mathematicae Gaph Theoy 28 ( A NOTE ON DOMINATION PARAMETERS IN RANDOM GRAPHS Athoy Boato Depatmet of Mathematics Wilfid Lauie Uivesity Wateloo, ON, Caada, N2L 3C5 aboato@oges.com ad Chagpig Wag Depatmet of Mathematics Ryeso Uivesity Tooto, ON, Caada, M5B 2K3 cpwag@yeso.ca Abstact Domiatio paametes i adom gaphs G(, p, whee p is a fixed eal umbe i (0, 1, ae ivestigated. We show that with pobability tedig to 1 as, the total ad idepedet domiatio umbes cocetate o the domiatio umbe of G(, p. Keywods: domiatio, adom gaphs, idepedet domiatio, total domiatio Mathematics Subect Classificatio: 05C69, 05C Itoductio Domiatio is a cetal topic i gaph theoy, with a umbe of applicatios i compute sciece ad egieeig. A set S of vetices i a gaph G is a domiatig set of G if each vetex ot i S is oied to some vetex of S. The authos gatefully ackowledge suppot fom NSERC ad MITACS.

2 336 A. Boato ad C. Wag The domiatio umbe γ(g is the miimum cadiality of a domiatig set of G. The cocetatio of the domiatio umbe of adom gaphs G(, p was ivestigated i [8]. Othe cotibutios to domiatio i adom gaph theoy iclude [2, 6, 7]. Fo backgoud o adom gaphs ad domiatio, the eade is diected to [1, 5] ad [3, 4], espectively. We say that a evet holds asymptotically almost suely (a.a.s. if the pobability that it holds teds to 1 as teds to ifiity. All logaithms ae i base e uless othewise stated, ad we use the otatio L = log 1/(1 p. Theoem 1 ([8]. Fo p (0, 1 fixed, a.a.s. γ(g(, p equals L L((L(log + 1 o L L((L(log + 2. Despite the fact that detemiistic gaphs of ode may have domiatio umbe equallig Θ( (such as a path P with γ(p = /3, Theoem 1 demostates that a.a.s. G(, p has domiatio umbe equallig (1 + o(1l = Θ(log. A set S is said to be a idepedet domiatig set of G if S is both a idepedet set ad a domiatig set of G (that is, S is a maximal idepedet set. A total domiatig set S i a gaph G is a subset of V (G satisfyig that evey v V (G is oied to at least oe vetex i S. The idepedet domiatio umbe of G, witte γ i (G, is the miimum ode of a idepedet domiatig set of G; the total domiatio umbe, witte γ t (G, is defied aalogously. It is staightfowad to see that γ(g γ i (G ad γ(g γ t (G. Howeve, the domiatio umbe may be of much smalle ode tha eithe the idepedet o total domiatio umbes; see fo example, [3, 4]. As poved i [9], thee ae cubic gaphs whee the diffeece betwee γ i ad γ is Θ(. Ou goal i this ote is to demostate that i G(, p with p fixed, asymptotically the idepedet ad total domiatio umbes cocetate o (1 + o(1l. I paticula, we pove the followig theoems. Theoem 2. A.a.s. γ t (G(, p equals L L((L(log + 1 o L L((L(log + 2. Theoem 3. A.a.s. we have that L L((L(log + 1 γ i (G(, p L.

3 A Note o Domiatio Paametes i Radom Gaphs 337 As the poofs of the theoems ae techical though elemetay we peset them i the ext sectio. Fo both poofs, we compute the asymptotic expected value of each domiatio paamete, ad the aalyze its vaiace. The secod momet method (see Chapte 4 of [1], fo example completes the poofs. All gaphs we coside ae fiite, udiected, ad simple. If A is a evet i a pobability space, the we wite P(A fo the pobability of A i the space. We use the otatio E(X ad V a(x fo the expected value ad vaiace of a adom vaiable X o G(, p, espectively. Thoughout, is a positive itege, all asymptotics ae as, ad p (0, 1 is a fixed eal umbe. 2. Poofs of Theoems 2 ad 3 The poofs ae peseted i the followig two subsectios. We ote the followig facts fom [8]. Fo 1, let X be the umbe of domiatig sets of size. Fix a -set S 1. Deote by S( the set of -sets which itesect S 1 i elemets. Let I 1 ad I be idicato adom vaiables, whee the evets I 1 = 1 ad I = 1 epeset that S 1 ad S S( ae domiatig sets, espectively. Let A = ( 1 =0 ( ( E(I 1 I. Lemma 4 ([8]. The adom vaiable X satisfies the followig popeties. (1 E(X = ( (1 (1 p. (2 Fo L L((L(log +2, we have that E (X as. (3 Fo L L((L(log + 2, whee A E 2 (X (1 + 2(1 p 2 (1 + o(1 + g(1 (2.1 g(1 = 2 1 ( 1! exp ( (1 p 2 1 2(1 p. (,

4 338 A. Boato ad C. Wag 2.1. Poof of Theoem 2 Fo a positive itege, the adom vaiable X t deotes the umbe of total domiatig sets of size. By Chebyshev s iequality, the poof of the theoem will follow oce we show that E(X t as, ad V a(x t = o(e2 (X t. (See, fo example, Sectio 4.3 of [1]. Lemma 5. If = L L((L(log + 2, the ( E(X t = (1 (1 p (1 (1 p 1 (1 + o(1. P oof. Fo 1 (, deote by E the evet that the subgaph iduced by a give -set S has o isolated vetices. We have that E(X t = ( (1 (1 p P(E. It is ot had to show that fo all, P(E 1 (1 p 1, ad so lim P(E = 1. The poof follows sice fo = L L((L(log +2, lim (1 (1 p 1 = 1. We ext show that fo a cetai value of, the expected value of X t cocetates o the expected value of X. Lemma 6. If = L L((L(log +2, the E(X t = (1+o(1E(X. P oof. By Lemmas 4 ad 5, we have that E(X t E(X = (1 (1 p 1 (1 + o(1. Hece, E(X lim t E(X = lim (1 (1 p 1 (1 + o(1 = 1. By Lemmas 4 ad 6, the poof of the followig lemma is immediate. Lemma 7. If = L L((L(log + 2, the E(X t as.

5 A Note o Domiatio Paametes i Radom Gaphs 339 We ow aalyze the vaiace of the adom vaiable X t. Lemma 8. If = L L((L(log + 2, the V a(x t = o(e 2 (X t. P oof. Fo 1 (, let I be the coespodig idicato adom vaiables. Hece, ( X t = I. =1 By the lieaity of expectatio, we have that (2.2 E((X t 2 = ( ( (I E 2 ( + 2 E ( I i I =1 i i = E ( X t ( + 2 E ( I i I. We fix a -set S 1. Fo 0 1, deote by S( the set of -sets which itesect S 1 i elemets. Let I t 1 ad It be the idicato adom vaiables, whee the evets I t 1 = 1 ad It = 1 epeset that S 1 ad S S( ae total domiatig sets, espectively. The ( 2 E(I i I = i ( Togethe with (2.2, we obtai that 1 =0 ( ( E(I t 1I t. (2.3 E((X t 2 = E(X t + ( 1 ( ( =0 E(I t 1 I t = E ( X t + A t, whee A t = ( 1 ( ( =0 E(I t 1 I t. As each total domiatig set is a domiatig set, A t A. By Lemmas 4 ad 6, we theefoe have that ( (2.4 A t g(1 whee g(1 is give i ( (1 + o(1e 2 (X t (1 + 2(1 p 2,

6 340 A. Boato ad C. Wag By (2.3 ad (2.4 we have that (2.5 V a(x t E 2 (X t 1 E(X t + (1 + o(1 (2(1 p 2 + g(1( E 2 (X t. To show that V a(x t = o(e 2 (X, t it suffices by Lemma 7 to show that g(1 ( E 2 (X t = o(1. Fo sufficietly lage we have that g(1 ( E 2 (X t = 2 1 ( 1! exp ( ( (1 p 2 1 2(1 p ( ( (1 (1 p (1 (1 p 1 2 (1 + o(1 33 ( 1 2(1 p + (1 p 2 1 (1 (1 p 2 2 (1 (1 p 1 2 (1 + o(1 33 p(1 p2 1 (1+ (1 (1 p 2 (1+ 2p(1 p 1 (1 (1 p 1 2 (1 + o(1, whee the fist equality follows by (2.1 ad sice exp(x 1 + x if x is close to 0. Sice 1 + x exp(x fo x 0, we obtai that g(1 ( E 2 (X t ( 33 ( p(1 p 2 1 exp 2p(1 p 1 (1 (1 p 2 + (1 (1 p 1 2 (1 + o(1 ( 33 exp (L(log 2 (1 + o(1p (1 + o(1 = o(1, as = L L((L(log Poof of Theoem 3 We use the followig lemma, whose poof is staightfowad ad so is omitted. Fo 1, let X I be the adom vaiable which deotes the umbe of idepedet domiatig sets of size.

7 A Note o Domiatio Paametes i Radom Gaphs 341 Lemma 9. (1 Fo all 1 ( E(X I = (1 (1 p (1 p ( 2. (2 Let λ ( 1 2, 1 be fixed. Fo L + 1 2λL, as we have that E(X I. Ou fial lemma estimates the vaiace of X I. Lemma 10. Let 2λL, p (0, 1 ad λ ( 1 2, 1 be fixed. Fo L + 1 V a ( ( X I (log = E 2 (X I 4 O. 1 λ By Chebyshev s iequality ad Lemmas 9 ad 10, we have that P(γ i (G > = P(X I = 0 P( X I E(X I E(X I V a(xi E 2 (X I = o(1. The assetio of Theoem 3 follows, theefoe, oce we pove Lemma 10. Poof of Lemma 10. We deote by E((X I2 the expectatio of the umbe of odeed pais of idepedet domiatio sets of size i G G(, p. The expectatio satisfies ( (X E I 2 = ( ( ( (1 (1 p 2( 2+ (2.6 ( 1 (1 p 2( (1 p 2( 2 ( 2. The explaatios fo the tems i the equatio (2.6 ae as follows. The vetices of the fist idepedet domiatig set S 1 may be chose i ( ways. The idepedet domiatig sets S 1 ad S 2 may have elemets i commo. These vetices may be chose i ( ways. The est of vetices of S 2 may have to be chose fom V (G\S 1, which gives the ( tem. Evey vetex ot i S 1 S 2 must be oied to oe of S 1 ad oe of S 2, ad so we obtai the tem (1 (1 p 2( 2+. Evey vetex i S 1 \S 2

8 342 A. Boato ad C. Wag must be oied to oe of S 2 \S 1, ad evey vetex i S 2 \S 1 must be oied to oe of S 1 \S 2, ad so we have the tem (1 (1 p 2(. Both sets S 1 ad S 2 ae idepedet, which supplies the last tem. Obseve that (1 p (1 p. Hece, by (2.6 ad Lemma 9 (1, we have that (2.7 ( (X E I 2 E 2 ( X I 1 ( ( ( ( =2 ( ( (1 p ( 2. By the choice of it follows that (( ( 1 ( + 1 (2.8 ( ( = (1 2 (log O 2 ( + 2 (log 3 + O 2 ( (log 4 = 1 + O, 2 ad (2.9 1 ( =2 ( ( (1 p (2 = O ( (log 4 1 λ. By (2.7, (2.8, ad (2.9 the assetio follows. Refeeces [1] N. Alo ad J. Spece, The Pobabilistic Method (Wiley, New Yok, [2] P.A. Deye, Applicatios ad vaiatios of domiatio i gaphs, Ph.D. Dissetatio, Depatmet of Mathematics (Rutges Uivesity, [3] T.W. Hayes, S.T. Hedetiemi ad P.J. Slate, Fudametals of Domiatio i Gaphs (Macel Dekke, New Yok, [4] T.W. Hayes, S.T. Hedetiemi ad P.J. Slate (eds., Domiatio i Gaphs: Advaced Topics (Macel Dekke, New Yok, 1998.

9 A Note o Domiatio Paametes i Radom Gaphs 343 [5] S. Jaso, T. Luczak ad A. Ruciński, Radom Gaphs (Joh Wiley ad Sos, New Yok, [6] C. Kaise ad K. Webe, Degees ad domiatio umbe of adom gaphs i the -cube, Rostock. Math. Kolloq. 28 ( [7] K. Webe, Domiatio umbe fo almost evey gaph, Rostock. Math. Kolloq. 16 ( [8] B. Wielad ad A.P. Godbole, O the domiatio umbe of a adom gaph, The Electoic Joual of Combiatoics 8 (2001 #R37. [9] I.E. Zveovich ad V.E. Zveovich, The domiatio paametes of cubic gaphs, Gaphs ad Combiatoics 21 ( Received 11 Jauay 2008 Revised 3 Mach 2008 Accepted 3 Mach 2008