Strong Complementary Acyclic Domination of a Graph

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1 Aals of Pure ad Applied Mathematics Vol 8, No, 04, ISSN: X (P), (olie) Published o 7 December 04 wwwresearchmathsciorg Aals of Strog Complemetary Acyclic Domiatio of a Graph NSaradha ad VSwamiatha Departmet of Mathematics, SCSVMV Uiversity, Kachipuram, Idia saaradha@yahoocom Ramauja Research Ceter i Mathematics, Saraswathi Narayaa College Madurai, Idia swamiathasulaesri@gmailcom Received 8 September 04; accepted November 04 Abract Let G = (V,E) be a graph A domiatig set S of G is called a rog complemetary acyclic domiatig set if S is a rog domiatig set ad the iduced subgraph V - S is acyclic The miimum cardiality of a rog complemetary acyclic domiatig set of G is called the rog complemetary acyclic domiatio umber of G ad is deoted by ( I this paper, we itroduce ad discuss the cocept of rog complemetary acyclic domiatio umber of GWe determie this umber for some adard graphs ad obtai some bouds for geeral graphs Its relatioship with other graph theoretical parameters are also iveigated Keywords: Domiatio umber, rog domiatio umber, rog complemetary acyclic domiatio umber AMS Mathematics Subject Classificatio (00): 05C35 Itroductio By a graph we mea a fiite, simple, ad udirected graph G(V,E) where V deotes its vertex set ad E its edge set Uless otherwise ated, the graph G has vertices ad e edges Degree of a vertex v is deoted by d(v), the maximum degree of a graph G is deoted by ( We deote a cycle o vertices by C, a path o vertices by P, ad a complete graph o vertices by K A graph G is coected if ay two vertices of G are coected by a path A maximal coected sub graph of a graph G is called a compoet of G The umber of compoets of G is deoted by ω ( The complemet G of G is the graph with vertex set V i which two vertices are adjacet if ad oly if they are ot adjacet i G A graph G is said to be acyclic if it has o cycles A tree is a coected acyclic graph A bipartite graph is a graph whose vertex set ca be partitioed ito two disjoit sets V ad V such that every edge has oe ed i V ad aother ed i V A Complete bipartite graph is a bipartite graph where every vertex of V is adjacet to every vertex i V The Complete bipartite graph with partitios of order V = m ad V =, deoted by K m, A ar deoted by K, is a tree with oe root vertex ad - pedat vertices A biar, deoted by D(r,s) is the graph obtaied by joiig the root 83

2 vertices of the ars K, r NSaradha ad VSwamiatha K, ad s A wheel graph deoted by W is a graph with vertices formed by joiig a sigle vertex to all vertices of C A helm graph, deoted by H is a graph obtaied from the wheel W by attachig a pedat vertex to each vertex i the outer cycle of W Coroa of two graphs G ad G, deoted by G G is the graph obtaied by takig oe copy of G ad V(G ) copies of G i which i th vertex of G is joied to every vertex i the ith copy of G If S is a subset of V, the S deotes the vertex iduced sub graph of G iduced by S The ope eighborhood of a set S of vertices of graph G, deoted by N(S) is the set of all vertices adjacet to some vertex i S, ad N ( S) S is called the closed eighbourhood of S, deoted by N[S] The diameter of a coected graph is the maximum diace betwee two vertices i G ad is deoted by diam( A cut-vertex of a graph G is a vertex whose removal icreases the umber of compoets A vertex cut of a coected graph G is a set of vertices whose removal results i a discoected graph The coectivity or vertex coectivity of a graph G, deoted by k((where G is ot complete) is the size of a smalle vertex cut A coected sub graph H of a coected graph G is called a H-cut if ω ( G H ) The chromotic umber of a graph G, deoted by χ ( is the miimum umber of colors eeded to color all the vertices a graph G i which adjacet vertices receive diict colors For ay real umber x deotes the large iteger less tha or equal to x A Nordhaus-Gaddum type result is a lower or upper boud o the sum or product of a parameter of a graph ad its complemet Terms ot defied here are used i the sese of [] A subset of V is called a domiatig set of G if every vertex i V-S is adjacet to at lea oe vertex i S The domiatio umber ( of G is the miimum cardiality take over all domiatig sets i GA domiatig set S of G is called a rog domiatig set of G if for every v V S there exi a vertex u S such that uv E( ad d( u) d ( v) The miimum cardiality take over all rog domiatig sets is the rog domiatio umber ad is deoted by ( A domiatig set S of G is called a complemetary acyclic domiatig set of G if V - S is acyclicthe miimum cardiality take over all complemetary acyclic domiatig sets is the complemetary acyclic domiatio umber ad is deoted by ( May authors have itroduced differet types of domiatio parameters by imposig coditios o the domiatig set []The cocept of rog domiatio has bee itroduced by Sampathkumar ad Pushpalatha [5] I this paper, we use this idea to develop the cocept of rog complemetary acyclic domiatio umber of a graph Strog complemetary acyclic domiatio Defiitio A domiatig set S of G is called a rog complemetary acyclic domiatig set if S is a rog domiatig set ad the iduced subgraph V - S is 84

3 Strog Complemetary Acyclic Domiatio of a Graph acyclic The miimum cardiality of a rog complemetary acyclic domiatig set of G is called the rog complemetary acyclic domiatio umber of G ad is deoted by ( Example A Strog complemetary acyclic domiatig set of a graph G is give below: v v v 6 v 3 v 7 v 4 v 5 Figure : G {v 3,v 5 } is a rog complemetary acyclic domiatig set of G For ay graph G, V( is a rog complemetary acyclic domiatig set Remark 3 Throughout this paper we cosider oly graphs for which rog complemetary acyclic domiatig set exis The complemet of the rog complemetary acyclic domiatig set eed ot be a rog complemetary acyclic domiatig set Example 4 v v v 4 v 3 v 5 v 6 v 7 v 8 v 9 v 0 v v Figure : G {v,v,v 3,v 4 } is a rog complemetary acyclic domiatig set But its complemet is ot a rog complemetary acyclic domiatig set Defiitio 5 A Strog Complemetary acyclic domiatig set S of G is miimal if o proper subset of S is a Strog Complemetary acyclic domiatig set of G Remark 6 Ay superset of rog complemetary acyclic domiatig set of G is also a rog complemetary acyclic domiatig set of G Sice if S is a rog complemetary acyclic domiatig set of G ad u V S S u is a rog complemetary, the { } 85

4 NSaradha ad VSwamiatha acyclic domiatig set of G Therefore rog complemetary acyclic domiatio is super hereditary Remark 7 A rog complemetary acyclic domiatig set of G is miimal iff it is - miimal Theorem 8 A rog complemetary acyclic domiatig set of G is miimal if ad oly if for each vertex u S oe of the followig coditios holds: u has a rog private eighbor i V S ( V S) { u} cotais a cycle Proof: Let S be a rog complemetary acyclic domiatig set of G Suppose S is miimal Let S S u is ot a rog complemetary acyclic domiatig set u The { } of G Therefore V S) { u} ( cotais a cycle or u has a rog private eighbour i V- S with respect to S Coversely, suppose for every u i S, oe of the coditios holds If () hods, the S { u} is ot a rog domiatig set If () holds, the S { u} is ot a complemetary acyclic Therefore, S is a miimal rog complemetary acyclic domiatig set of G Remark 9 Every rog complemetary acyclic domiatig set is a domiatig set But every domiatig set eed ot be a rog complemetary acyclic domiatig set Example 0 v v v 4 v 3 Figure 3: G 3 Here {v,v } is a rog complemetary acyclic domiatig set ad also a domiatig set Also {v } is a domiatig set but it is ot a rog complemetary acyclic domiatig set Theorem For ay graph G, s c a t ad the bouds are sharp Let S be a miimum rog complemetary acyclic domiatig set of G Let v V S The there exis u S such that u ad v are adjacet ad deg( u) deg( v) Therefore S is a rog domiatig set of G ad hece S is a domiatig set of G Therefore 86

5 Example Strog Complemetary Acyclic Domiatio of a Graph v v v 3 v 4 v 5 v 6 v 7 v 8 Figure 4: G 4 Here = = c a = 4 Theorem 3 ( K ) = K ( K, ) = ( ) =, 3 3 ( D, ) = r s 4 ( W ) = c a K m = 5 (, ) mi{ m, } Theorem 4 For ay path P m ( P m ) = if m = 3, N = if m = 3 or 3, N Proof: Case (i) Let G = P3, N Let v,v,v 3, v 3 be the vertices of V(P 3 ) {v,v 5, v 8 v 3- } is the uique rog domiatig set of P 3 It is also the rog complemetary acyclic domiatig set of P 3 Therefore, for all N ( P 3 ) = Case (ii) Let G = P3, N Let{v,v,v 3, v 3,v 3 } S ={v,v 5, v 8 v 3-,v 3 } ad S ={v,v 3,v 6, v 9 v 3 }are two rog complemetary acyclic domiatig sets of G Now S {, v, v, v } v = Also = v { v, v, v } = S = v Therefore, ( 3 ) P Also ( ) P = ad by Theorem 0, we have 3 ( ) P3 = Hece s t 87

6 NSaradha ad VSwamiatha Case (iii) Let G = P3, N Let{v,v,v 3, v 3,v 3,v 3 } S={v,v 5, v 8 v 3-,v 3`} is a rog complemetary acyclic domiatig ses of G Now S {, v, v, v } v = = v Therefore ) ( P3 ( P3 ) ( ) P3 = Also ad by Theorem 0 Hece Theorem 5 ( ) = if C m m = 3, N = if m = 3 or 3, N Proof: The proof follows from Theorem Observatio 6 If a spaig sub graph H of a graph G has a rog complemetary acyclic domiatig set the G has a rog complemetary acyclic domiatig set Observatio 7 Let G be a coected graph ad H be a spaig sub graph of G If H has a rog complemetary acyclic domiatig set, the ( H ) ad the bouds are sharp Theorem 8 For ay coected graph G, with 3 vertices, ad the bouds are sharp Proof: The lower ad upper bouds follows from defiitio For K, the lower boud is attaied ad for K 4, the upper boud is attaied Observatio 9 For ay coected graph G with 3 vertices, = if ad oly if G P 4,C3 The Nordhaus-Gaddum type result is give below Theorem 0 Let G be a graph such that G ad G o isolates of order 3 The ( G ) 4 ad ( G ) ( ) Proof: The boud directly follows from Theorem 5 Relatioship with other graph theoretical parameters Theorem For ay coected graph with 3 vertices k( 3 ad the boud is sharp if ad oly if G K Proof: Let G be a coected graph with verticeswe kow that k ad by theorem 8, Hece k( 3 Suppose G is isomorphic K The clearly k( = 3 Coversely, let s t 88

7 Strog Complemetary Acyclic Domiatio of a Graph k( = 3 This is possible oly if = ad k = But k = ad so G K for which c a = Hece G K Theorem For ay coected graph G with 3 vertices, χ( ad the boud is sharp if ad oly if G K Proof: Let G be a coected graph with vertices We kow that χ ad by Theorem 8, Hece χ( Suppose G is isomorphic to K The clearly χ( = Coversely, let χ( = This is possible oly if = ad χ ( G ) = Sice χ ( G ) =, G is isomorphic to K for which = Hece G K Theorem 3 For ay coected graph G with 3 vertices, ( 3 ad the boud is sharp Proof: Let G be a coected graph with vertices, ( G ) ad by Theorem 8, Hece ( 3For K 5 the boud is sharp 3 Coclusio We foud rog complemetary acyclic domiatio umber for some adard graphs ad obtaied some bouds for geeral graphs Its relatioship with other graph theoretical parameters are also iveigated REFERENCES JABody ad USRMurthy, Graph Theory with Applicatios, the Macmilla Press Ltd, 976 TWHayes,STHedeteimi ad PJSlater (eds), Fudametals of Domiatios i Graphs, Marcel Dekker, New York, OOre, Theory of graphs, Amer Math Soc Colloq Publ, 38, Providece, 96 4 VRKulli ad BJaakiram, The o split domiatio umber of a graph, Idia J Pure Appl Math, 3(5) (000) ESampathkumar ad LPushpa Latha, Strog weak domiatio ad domiatio balace i a graph, Discrete Math, 6 (996) BJaakiram, NDSoer ad MADavis, Complemetary acyclic domiatio i graphs, J Idia Math Soc (NS), 7 (-4) (004) -6 7 SMuthammai, MBhaumathi ad PVidhya, Complemetary tree domiatio umber of a graph, It Mathematical Forum, 6 (6) (0) AKSiha, ARaa ad APal, The -tuple domiatio problem o trapezoid graphs, Aals of Pure ad Applied Mathematics, 7() (04) MPal, Itersectio graphs: a itroductio, Aals of Pure ad Applied Mathematics, 4() (03)