BOUNDS FOR OUT DEGREE EQUITABLE DOMINATION NUMBERS IN GRAPHS

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1 BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p), ISSN (o) Vol. 3(2013), Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA LUKA ISSN (o), ISSN X (p) BOUNDS FOR OUT DEGREE EQUITABLE DOMINATION NUMBERS IN GRAPHS Kuppusamy Markada Dharmaligam Abstract. Let G = (V, E) be a simple graph. Let D be a domiatig set ad u D. The edges from u to V D are called domiatig edges. A domiatig set D is called outdegree equitable if the differece betwee the cardialities of the sets of domiatig edges from ay two poits of D is at most oe. The miimum cardiality of a out degree equitable domiatig set is called the outdegree equitable domiatio umber ad is deoted by γ e.the existece of a outdegree equitable domiatig set is guarateed. Out degree equitable domiatio is itroduced i this paper. Miimum, miimal,idepedet outdegree equitable domiatig sets,out degree Equitable poits ad Equitable eighbourhood umbers are defied ad also obtaied the bouds for γ e. 1. Itroductio Prof. E. Sampathkumar itroduced the cocept of equitability i terms of outward ifluece. I a orgaizatio, the executive body which takes all decisios is to be formed o the basis of two criteria. (i) Members of the body should have cotacts with all the members of the orgaizatio. (ii) The iflueces of members of the executive body over the rest of the orgaizatio are to be equitable. The first criterio is take care of by the cocept of domiatio i graphs. The secod criterio is take care of by the ew cocept of Prof. E. Sampathkumar, amely outward equitability. A graph model for this is suggested as follows: Let S 1 ad S 2 be two subsets of the vertex set V of a graph G = (V, E). Members of S 1 are said to be outdegree equitable with respect to S 2 if they have equitable umber of eighbours i S 2.This 2010 Mathematics Subject Classificatio. 05C. Key words ad phrases. Upper ad lower Bouds for Out degree equitable domiatio Number. 149

2 150 K. M. DHARMALINGAM yields a ew parameter amely outdegree equitable domiatio umber.a study of this cocept ad the parameter is made i this paper. 2. Out Degree Equitable Domiatig sets - Miimal ad Miimum Outdegee Equitable Domiatig sets Defiitio 2.1. Let G = (V, E) be a simple graph. Let D be a domiatig set ad u D. The edges from u to V D are called domiatig edges. A domiatig set D is called outdegree equitable if the differece betwee the cardialities of the sets of domiatig edges from ay two poits of D is at most oe. The miimum cardiality of a outdegree equitable domiatig set is called the outdegree equitable domiatio umber ad is deoted by γ e.the existece of a outdegree equitable domiatig set is guarateed. Defiitio 2.2. A subset D of V is a miimal outdegree equitable domiatig set if o proper subset of D is a outdegree equitable domiatig set. Defiitio 2.3. A outdegree equitable domiatig set is said to be 1-miimal if D v is ot a outdegree domiatig set for all v D. Example Fig. 1.1 {1} is a outdegree equitable domiatig set of K 3 {2, 3} is a outdegree equitable domiatig set of K 3 {2, 3} is ot a miimal outdegree equitable domiatig set, sice {2, 3} cotais {2} which is a outdegree equitable domiatig set of K 3. Example Fig {1} is a outdegree equitable domiatig set. {1,2} is ot a outdegree equitable domiatig set. {1,2,3} is a outdegree equitable domiatig Set. Thus {1,2,3} is a outdegree equitable domiatig set for which the sub set {1, 2} is ot a outdegree equitable domiatig set but the subset {1} is a outdegree equitable domiatig set. Thus if outdegree equitable domiatig set D is 1-miimal, the D eed ot be miimal.

3 BOUNDS FOR OUT DEGREE EQUITABLE DOMINATION NUMBERS IN GRAPHS 151 Defiitio 2.4. Let D be a subset of V. Let u D. The out degree of u is defied as the umber of edges from u to V D (i.e) out degree of u is N(u) (V D). The out degree of u i D is deoted by d o D (u). The out degree of D deoted by D 0 is defied as D 0 = mi {do D (u)}. Defiitio 2.5. Let D be a subset of v. Let u 1, u 2 D. u 1 ad u 2 are said to be outdegree equitable poits (or simply equitable poits) if d o D (u 1) d o D (u 2) 1. Otherwise u 1 ad u 2 are said to be o-equitable poits. 3. Bouds for Outdegree Equitable Domiatio umber Theorem 3.1. Let G be a graph cotaiig two poits u, v with N(u), N(v) ϕ ad N[u] N[v] = ϕ. The 1 γ e (G) 2. Proof. Let D = V {u, v}. Sice N[u] N[v] = ϕ, u ad v are ot adjacet. Sice N(u) ϕ, N(v) ϕ ad N[u] N[v] = ϕ there exist distict vertices x, y i D such that x is adjacet to u ad y is adjacet to v. The out degree of ay poit i D is either 0 or 1. Hece D is a equitable domiatig set of G. Therefore γ e (G) 2. Theorem 3.2. Let D be a outdegree equitable domiatig set. The d(u) = V D if ad oly if (i) D is idepedet ad (ii) For every u V D there exists a uique vertex v D such that N(u) D = {v}. Proof. Suppose (i) ad (ii) hold. The clearly V D = d(u). Coversely, suppose V D = d(u). Suppose D is ot idepedet. The there exists u, v D such that u ad v are adjacet. The d(u) exceeds V D by at least two, which is a cotradictio. Therefore D is idepedet. Suppose (ii) is false. The N(u) D 2 for some u V D. Let v, w D such that v, w N(u). The d(x)exceeds V D by at least oe because u is couted x D twice, oce i d(v) ad i d(w), a cotradictio. Therefore (ii) must hold. Defiitio 3.1. Let D be a outdegree equitable domiatig set. The the outdegree of ay poit of D is either k or k + 1 where k is a o egative iteger. The outdegree of D deoted by d 0 is defied as the miimum of the outdegrees of vertices of D amely k. Corollary 3.1. Let D be a outdegree equitable domiatig set such that V D = d(u). Let d o be the outdegree of D the D Proof. V D = d(u) = D D (d 0 ). Therefore D d o D (u) D (d 0) d. 0+1 d o +1

4 152 K. M. DHARMALINGAM Corollary 3.2. Let D be a outdegree equitable domiatig set such that V D = d(u). Let k be the umber of poits i D with outdegree d 0. The γ e +k 2+d 0. Proof. V D = d(u) D = kd 0 + ( D k)(d 0 + 1) = kd 0 + D d 0 + D kd 0 k = D d o + D k + k = D (2 + d 0 ) D = + k 2 + d 0. Therefore γ e D = +k 2+d 0. Corollary 3.3. Let D be a outdegree equitable domiatig set such that V D = d(u). Number of edges from D to V D is D (d 0 + 1) k where k is the umber of vertices i D havig outdegree d 0. Proof follows from Corollary 3.2. Corollary 3.4. Let D be a outdegree equitable domiatig set such that V D = d(u). Number of edges of G ( D )( D 1) 2 + D (d 0 + 1) k ad maximum is attaied whe < V D > is complete. Proof is obvious. Corollary 3.5. If G has o isolates ad V D = d(u) for some outdegree equitable domiatig set D, the V D is a 1- miimal outdegree equitable domiatig set of G. Proof. Sice G has o isolates ad V D = d(u) we get that every poit of D is adjacet to some poit of V D. Therefore V D is a domiatig set. Also for every u V D, there exists a uique vertex v D such that N(u) D = {v}. Therefore V D is a out degree equitable domiatig set. Therefore 1-miimal out degree equitable domiatig set we get that V D is a 1-miimal out degree equitable domiatig set. [ Theorem 3.3. Let G be a graph of order such that γ e (G) = + 1 divides. +1 ]. The

5 BOUNDS FOR OUT DEGREE EQUITABLE DOMINATION NUMBERS IN GRAPHS 153 Proof. Let S be a miimum out degree equitable domiatig set of G. Suppose S is ot idepedet, the by previous theorem, V S < u S d(u) S = γ e (G) γ e < γ e < γ e ( + 1) + 1 < γ e (i.e) γ e (G) > + 1 which is a cotradictio. Therefore S is idepedet. Claim For every vertex u V S, there is a uique vertex v S such that N(u) S = {v}. Suppose this is ot true. The proceedig as above, we get a cotradictio. Therefore the claim holds. Therefore from a earlier propositio, V S = d(u). u S Claim For every vertex u S, d(u) =. Suppose ot. The there exists a vertex w S such that d(w) < Therefore d(u) < S. u S V S < S γ e < γ e < γ e ( + 1) γ e > + 1 which is a cotradictio. Therefore every vertex of S is of degree. Therefore V S = d(u) u S γ e = S γ e = γ e = γ e ( + 1). Therefore ( + 1) divides. Remark 3.1. Coverse of the above theorem is false. Example 3.1. Cosider the followig graph.

6 154 K. M. DHARMALINGAM Fig. 1.3 Here = 3, = 8, γ e = 4. That is ( + 1) divides. γ e +1. Refereces +1 = 8/4 = 2. Therefore [1] R. B. Alla ad R. C. Laskar,O domiatio ad idepedet domiatio umbers of a graph, Discrete Math., 23 (1978), [2] C. Berge, ad P. Duchet, Recet problem ad results about kerels i directed graphs, Discrete Math., 86 (1990), [3] A. P. Burger, E. J. Cockaye ad C. M. Myhardt, Domiatio Numbers for the Quee s graph, Bull. Ist. Comb. Appl., 10 (1994), [4] E. J. Cockaye, S. T. Hedetiemi, Towards a theory of domiatio i Graphs, Networks, 7 (1977), [5] E. J. Cockaye, R. M. Dawes ad S. T. Hedetiemi, Total domiatio i Graphs, 10 (1980), [6] E. J. Cockaye, C. W. Ko, ad F. B. Shepherd, Iequalities Cocerig domiatig sets i graphs, Techical Report DM-370-IR, Dept. Math., Uiv. Victoria,(1985). [7] Vekatasubramaia Swamiatha ad Kuppusamy Markada Dharmaligam Degree Equitable Domiatio o Graphs, Kragujevac J. Math., 35(1)(2011), [8] Kuppusamy Markada Dharmaligam Equitable Associate Graph of a Graph, Bull. Iter. Math. Virtual Ist., 2(1)(2012), Received by editors ; revised versio ; available olie Departmet of Mathematics, The Madura College, Madurai , Idia address: kmdharma6902@yahoo.i